Method of generating reference signal in wireless communication system

ABSTRACT

A method of generating a reference signal includes acquiring a base sequence and acquiring a reference signal sequence with a length N from the base sequence. Good PAPR/CM characteristics of the reference signal can be kept to enhance performance of data demodulation or uplink scheduling.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.12/913,654 (now U.S. Pat. No. 8,098,760) filed on Oct. 27, 2010, whichis a continuation of U.S. application Ser. No. 12/205,530 (now U.S. Pat.No. 7,848,448, issued on Dec. 7, 2010) filed on Sep. 5, 2008, whichclaims the benefit of priority from U.S. Provisional Application No.60/970,754 filed on Sep. 7, 2007, U.S. Provisional Application No.60/972,401 filed on Sep. 14, 2007, U.S. Provisional Application No.60/978,415 filed on Oct. 9, 2007, U.S. Provisional Application No.60/978,687 filed on Oct. 9, 2007, and Korean Patent Application No.10-2008-0033799 filed on Apr. 11, 2008, which are all incorporated byreference in their entirety herein.

BACKGROUND

1. Technical Field

The present invention relates to wireless communication and, moreparticularly, to a method of generating a reference signal in a wirelesscommunication system.

2. Related Art

In general, a sequence is used for various channels and signals in awireless communication system. The sequence in the wirelesscommunication system needs to satisfy the following characteristics:

(1) Good correlation characteristics to provide high detectionperformance,

(2) Low CM (Cubic Metric) to enhance efficiency of a power amplifier,

(3) Generation of a large number of sequences to transmit a large amountof information or to facilitate cell planning,

(4) Being able to be generated in a closed form in order to reduce acapacity of a memory for the sequence.

A downlink synchronization channel is used to perform time and frequencysynchronization between a base station and a user equipment and toperform cell searching. A downlink synchronization signal, namely, asequence, is transmitted on the downlink synchronization channel, andsynchronization is performed through a correlation operation with thereceived downlink synchronization signal. A physical cell ID can beidentified through the downlink synchronization channel. Because aunique cell ID should be identified, as the number of availablesequences is increased, it is advantageous in terms of cell planning.

An uplink synchronization channel is used to perform time and frequencysynchronization and to perform access for a network registration, ascheduling request, or the like. A sequence is transmitted on the uplinksynchronization channel, and each corresponding sequence is recognizedas a single opportunity. Upon detecting a sequence, the base station canrecognize through which opportunity the user equipment has transmittedthe uplink synchronization channel. In addition, through the detectedsequence, a timing tracking, a residual frequency offset, or the like,may be estimated. As the number of opportunities is increases,probability of collision between user equipments can be reduced. Thus, alarger number of sequences is advantageous in terms of cell planning.The uplink synchronization channel is called a random access channel(RACH) or a ranging channel depending on a system.

A sequence can be used as control information transmitted on a controlchannel. This means the control information such as an ACK(Acknowledgement)/NACK (Negative-Acknowledgement) signal, a CQI (ChannelQuality Indicator), etc. can be mapped to the sequence. The largernumber of available sequences is advantageous to transmit variouscontrol information.

A scrambling code is used to provide randomization or peak-to-averagepower ratio (PAPR) reduction. In terms of cell planning, a larger numberof sequences are advantageous to be used for scrambling codes.

When several users are multiplexed in a single channel through codedivision multiplexing (CDM), a sequence may be used to guaranteeorthogonality among users. A multiplexing capacity is related to thenumber of available sequences.

A reference signal is used by a receiver to estimate a fading channeland/or is used to demodulate data. Further, the reference signal is usedto obtain synchronization when the user equipment awakes from atime/frequency tracking or in sleep mode. In this manner, the referencesignal is utilized variably. The reference signal uses a sequence, andthe larger number of sequences is advantageous in terms of cellplanning. The reference signal is also called as pilot.

There are two types of uplink reference signals: a demodulationreference signal and a sounding reference signal. The demodulationreference signal is used for channel estimation for data demodulation,and the sounding reference signal is used for user scheduling. Inparticular, the uplink reference signal is transmitted by a userequipment with a limited battery capacity, so PAPR or CM characteristicsof the sequences used for the uplink reference signal are critical. Inaddition, in order to lower the cost of the user equipment, it isnecessary to reduce a mount of the memory required for generatingsequences.

SUMMARY

A method is sought for generating a sequence suitable for an uplinkreference signal.

A method is sought for transmitting an uplink reference signal.

In an aspect, a method of generating a reference signal in a wirelesscommunication system is provided. The method includes acquiring a basesequence x_(u)(n) and acquiring a reference signal sequence r(n) with alength N from the base sequence x_(u)(n), wherein the base sequencex_(u)(n) is expressed by x_(u)(n)=e^(jp(n)π/4), and if N=12, at leastone of the values provided in the below table is used as a value of thephase parameter p(n):

p(0), . . . , p(11) −1 3 −1 1 1 −3 −3 −1 −3 −3 3 −1 −1 3 1 3 1 −1 −1 3−3 −1 −3 −1 −1 −3 1 1 1 1 3 1 −1 1 −3 −1 −1 3 −3 3 −1 3 3 −3 3 3 −1 −1

Further, if N=24, at least one of the values provided in the below tablecan be used as a value of the phase parameter p(n):

p(0), . . . , p(23) −1 −3 3 −1 −1 −1 −1 1 1 −3 3 1 3 3 1 −1 1 −3 1 −3 11 −3 −1 −1 −3 3 −3 −3 −3 −1 −1 −3 −1 −3 3 1 3 −3 −1 3 −1 1 −1 3 −3 1 −1−1 −1 1 −3 1 3 −3 1 −1 −3 −1 3 1 3 1 −1 −3 −3 −1 −1 −3 −3 −3 −1 −1 −3 11 3 −3 1 1 −3 −1 −1 1 3 1 3 1 −1 3 1 1 −3 −1 −3 −1 −1 −3 −1 −1 1 −3 −1−1 1 −1 −3 1 1 −3 1 −3 −3 3 1 1 −1 3 −1 −1

The reference signal sequence r(n) can be acquired asr(n)=e^(jαn)x_(u)(n), by a cyclic shift α of the base sequence x_(u)(n).

In another aspect, a method for transmitting a reference signal in awireless communication system is provided. The method includes acquiringa reference signal sequence r(n) with a length N from a base sequencex_(u)(n), mapping the reference signal sequence to the N number ofsubcarriers, and transmitting the mapped reference signal sequences onan uplink channel,

wherein the base sequence x_(u)(n) is expressed by

x_(u)(n)=e^(jp(n)π/4), and if N=12, at least one of the values providedin the below table is used as a value of the phase parameter p(n):

p(0), . . . , p(11) −1 3 −1 1 1 −3 −3 −1 −3 −3 3 −1 −1 3 1 3 1 −1 −1 3−3 −1 −3 −1 −1 −3 1 1 1 1 3 1 −1 1 −3 −1 −1 3 −3 3 −1 3 3 −3 3 3 −1 −1

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic block diagram of a transmitter according to anembodiment of the present invention.

FIG. 2 is a schematic block diagram of a signal generator according toSC-FDMA scheme.

FIG. 3 shows the structure of a radio frame.

FIG. 4 is an exemplary view showing a resource grid for an uplink slot.

FIG. 5 shows the structure of an uplink sub-frame.

FIG. 6 is a conceptual view showing cyclic extension.

FIG. 7 shows a truncation method.

FIG. 8 is a flow chart illustrating the process of a reference signaltransmission method according to an embodiment of the present invention.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

Hereinafter, downlink refers to communication from a base station (BS)to a user equipment (UE), and uplink refers to communication from the UEto the BS. In the downlink, a transmitter may be a part of the BS and areceiver may be a part of the UE. In the uplink, a transmitter may be apart of the UE, and a receiver may be a part of the BS. The UE may be afixed or mobile, and may be referred to as another terminology, such asa mobile station (MS), a user terminal (UT), a subscriber station (SS),a wireless device, etc. The BS is generally a fixed station thatcommunicates with the UE and may be referred to as another terminology,such as a node-B, a base transceiver system (BTS), an access point, etc.There may be one or more cells within the coverage of the BS.

I. System

FIG. 1 is a schematic block diagram showing a transmitter according toan embodiment of the present invention.

Referring to FIG. 1, a transmitter 100 includes a reference signalgenerator 110, a data processor 120, a physical resource mapper 130 anda signal generator 140.

The reference signal generator 110 generates a sequence for a referencesignal. There are two types of reference signals: a demodulationreference signal and a sounding reference signal. The demodulationreference signal is used for channel estimation for data demodulation,and the sounding reference signal is used for uplink scheduling. Thesame reference signal sequence may be used for the demodulationreference signal and the sounding reference signal.

The data processor 120 processes user data to generate complex-valuedsymbols. The physical resource mapper 130 maps the complex-valuedsymbols for the reference signal sequence and/or user data to physicalresources. The complex-valued symbols may be mapped to mutuallyexclusive physical resources. The physical resources may be called asresource elements or subcarriers.

The signal generator 140 generates a time domain signal to betransmitted via a transmit antenna 190. The signal generator 140 maygenerate the time domain signal according to an single carrier-frequencydivision multiple access (SC-FDMA) scheme and, in this case, the timedomain signal outputted from the signal generator 140 is called anSC-FDMA symbol or an orthogonal frequency division multiple access(OFDMA) symbol.

In the following description, it is assumed that the signal generator140 uses the SC-FDMA scheme, but it is merely taken as an example andthere is no limit of the multi-access scheme to which the presentinvention is applied. For example, the present invention can be appliedfor various other multi-access schemes such as an OFDMA, code divisionmultiple access (CDMA), time division multiple access (TDMA) orfrequency division multiple access (FDMA).

FIG. 2 is a schematic block diagram of a signal generator according tothe SC-FDMA scheme.

With reference to FIG. 2, the signal generator 200 includes a discreteFourier transform (DFT) unit 220 to perform DFT, a subcarrier mapper230, and an inverse fast Fourier transform (IFFT) unit 240 to performIFFT. The DFT unit 220 performs DFT on inputted data and outputsfrequency domain symbols. The subcarrier mapper 230 maps the frequencydomain symbols to each subcarrier, and the IFFT unit 230 performs IFFTon inputted symbols to output a time domain signal.

A reference signal may be generated in the time domain and inputted tothe DFT unit 220. Alternatively, the reference signal may be generatedin the frequency domain and directly mapped to subcarriers.

FIG. 3 shows the structure of a radio frame.

With reference to FIG. 3, a radio frame includes ten subframes. Eachsubframe includes two slots. An interval for transmitting a singlesubframe is called a transmission time interval (TTI). For example, theTTI may be 1 milli-second (ms) and the interval of a single slot may be0.5 ms. A slot may include a plurality of SC-FDMA symbols in the timedomain and a plurality of resource blocks in the frequency domain.

The structure of the radio frame is merely an example, and the number ofsubframes included in the radio frame, the number of slots included inthe subframe, and the number of SC-FDMA symbols included in the slot mayvary.

FIG. 4 shows a resource grid for an uplink slot.

Referring to FIG. 4, an uplink slot includes a plurality of SC-FDMAsymbols in the time domain and a plurality of resource blocks in thefrequency domain. Here, it is shown that the uplink slot includes sevenSC-FDMA symbols, and a resource block includes twelve subcarriers, butthose are merely examples, and the present invention is not limitedthereto.

Each element of the resource grid is called a resource element. A singleresource block includes 12×7 resource elements. The number (N^(UL)) ofresources blocks included in the uplink slot depens on an uplinktransmission bandwidth.

FIG. 5 shows the structure of an uplink subframe.

With reference to FIG. 5, an uplink subframe may be divided into twoparts: a control region and a data region. A middle portion of thesubframe is allocated to the data region, and both side portions of thedata region are allocated to the control region. The control region is aregion for transmitting control signals, which is typically allocated toa control channel. The data region is a region for transmitting data,which is typically allocated to a data channel. A channel allocated tothe control region is called a physical uplink control channel (PUCCH),and a channel allocated to the data region is called a physical uplinkshared channel (PUSCH). A UE cannot simultaneously transmit the PUCCHand the PUSCH.

The control signal includes an ACK (Acknowledgement)/NACK(Negative-Acknowledgement) signal which is an hybrid automatic repeatrequest (HARD) feedback for downlink data, a channel quality indicator(CQI) indicating a downlink channel condition, a scheduling requestsignal which is used to request an uplink radio resource, or the like.

The PUCCH uses a single resource block that occupies mutually differentfrequencies in each of two slots of a subframe. Two resource blocksallocated to the PUCCH is frequency-hopped at a slot boundary. Here, itis illustrated that two PUCCHs, one having m=0 and another having m=1,are allocated to a subframe, but a plurality of PUCCHs may be allocatedto a subframe.

II. Zadoff-Chu (ZC) Sequence

A Zadoff-Chu (ZC) sequence is commonly used in a wireless communicationbecause it has good CM characteristics and correlation characteristics.The ZC sequence is one of constant amplitude and zero auto correlation(CAZAC) based sequences. The ZC sequence has idealistic characteristicswith a constant amplitude at both time and frequency domains through DFT(or IDFT) and a periodic auto-correlation in the form of impulse. Thus,application of the ZC sequence to DFT-based SC-FDMA or OFDMA shows verygood PAPR (or CM) characteristics.

A generating equation of a ZC sequence with a length of N_(ZC) is asfollows:

$\begin{matrix}{{x_{u}(m)} = \left\{ \begin{matrix}{\mathbb{e}}^{{- j}\frac{\pi\;{{um}{({m + 1})}}}{N_{ZC}}} & {{for}\mspace{14mu}{odd}\mspace{14mu} N_{Zc}} \\{\mathbb{e}}^{{- j}\frac{\pi\;{um}^{2}}{N_{ZC}}} & {{for}\mspace{14mu}{even}\mspace{14mu} N_{Zc}}\end{matrix} \right.} & \left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack\end{matrix}$

where 0≦m≦N_(ZC)−1, and ‘u’ denotes a root index which is a naturalnumber not larger than N_(ZC). The root index u is relatively prime withN_(ZC). It means that when N_(ZC) is determined, the number of rootindexes becomes the number of available root ZC sequences. Accordingly,when the N_(ZC) is a prime number, the largest number of root ZCsequences can be obtained. For example, if N_(ZC) is 12 which is acomposite number, the number of available root ZC sequences is 4 (u=1,5, 7, 11). If N_(ZC) is 13 which is a prime number, the number ofavailable root ZC sequences is 12 (u=1, 2, . . . , 10).

In general, a ZC sequence having the length of a prime number has betterCM or correlation characteristics than those of a ZC sequence having thelength of a composite number. Based on this fact, there are two methodsfor increasing the number of ZC sequences when the length of the ZCsequences desired to be generated is not a prime number: One is a methodbased on a cyclic extension and the other is a method based ontruncation.

FIG. 6 is a conceptual view showing the cyclic extension method. Thecyclic extension method refers to a method in which (1) when the lengthof desired ZC sequences is ‘N’, (2) the ZC sequences are generated byselecting a prime number smaller than the desired length N as N_(ZC),and (3) the generated ZC sequences are cyclically extended to theremaining portion (N-N_(ZC)) to generate ZC sequences with the length N.For example, if N is 12, N_(ZC) is selected to be 11 to obtain all the10 cyclic-extended ZC sequences.

By using the ZC sequence x_(u)(m) of Equation 1, the cyclic-extendedsequences r_(CE)(n) can be expressed as shown below:r _(CE)(n)=x _(u)(n mod N _(ZC))  [Equation 2]

where 0≦n≦N−1, ‘a mod b’ denotes a modulo operation, which means aresidual obtained by dividing ‘a’ by ‘b’, and N_(ZC) denotes the largestprime number among natural numbers not larger than N.

FIG. 7 is a conceptual view showing a truncation method. The truncationmethod refers to a method in which (1) when the length of desired ZCsequences is N, (2) a prime number larger than the desired length N isselected as N_(ZC) to generate ZC sequences, and (3) the remainingportion (N_(ZC)-N) is truncated to generate ZC sequences with the lengthN. For example, if N is 12, N_(ZC) is selected to be 13 to obtain allthe twelve truncated ZC sequences.

By using the ZC sequence x_(u)(m) of Equation 1, the truncated andgenerated sequences r_(TR)(n) can be expressed as shown below:r _(TR)(n)=x _(u)(n)  [Equation 3]

where 0≦n≦N−1, and N_(ZC) denotes the smallest prime number amongnatural numbers of not smaller than N.

When sequences are generated by using the above-described ZC sequences,the number of available sequences is maximized when N_(ZC) is a primenumber. For example, if the length N of desired sequences is 11, when ZCsequences of N_(ZC)=11 is generated, the number of available sequencesis a maximum 10. If the amount of required information or the number ofused sequences should be more than ten sequences, the ZC sequence cannotbe used.

If the length of desired sequences is N=12, N_(ZC)=11 is selected andthe cyclic extension is performed or N_(ZC)=13 is selected andtruncation is performed to thereby generate ten ZC sequences in case ofthe cyclic extension and twelve ZC sequences in case of the truncation.In this case, however, if more sequences are required (e.g., 30sequences), ZC sequences having such good characteristics as satisfyingthe sequences cannot be generated.

In particular, if sequences having good CM characteristics are required,the number of available sequences may be severely reduced. For example,preferably, sequences used for a reference signal is lower than a CMvalue in quadrature phase shift keying (QPSK) transmission when powerboosting is considered. When SC-FDMA scheme is used, a CM value in QPSKtransmission is 1.2 dB. If sequences satisfying the QPSK CM requirementsare selected from among the available ZC sequences, the number ofavailable sequences to be used for the reference signal would bereduced. In detail, the below table shows CM values of sequencesgenerated after being cyclic-extended by selecting N_(ZC)=1 in casewhere the length of desired sequences is N=12.

TABLE 1 Sequence index u CM [dB] 0 1 0.17 1 2 1.32 2 3 1.50 3 4 0.85 4 50.43 5 6 0.43 6 7 0.85 7 8 1.50 8 9 1.32 9 10 0.17

As noted in the above table, if a threshold value is 1.2 dB, therequirements of QPSK CM, the number of available sequences is reducedfrom ten to six (u=0, 4, 5, 6, 7, 10).

Therefore, a method of generating a sequence that may have good CM andcorrelation characteristics and can reduce the memory capacity requiredfor generating or storing available sequences is required.

III. Sequence Generating Equation

A closed-form generating equation for generating sequences having goodCM and correlation characteristics is a polynomial expression with auniform size and a k-th order phase component.

The closed-form generating equation with respect to a sequence r(n) isas follows:r(n)=x _(u)(n), 0≦n≦N−1, x _(u)(m)=e ^(−j(u) ⁰ ^(m) ^(k) ^(+u) ¹ ^(m)^(k-1) ^(+ . . . +u) ^(k-1) ^(m) ¹ ^(+u) ^(k) ⁾  [Equation 4]

where m=0, 1, . . . , N−1, ‘N’ denotes the length of the sequence r(n),and u₀, u₁, . . . , u_(k) denote arbitrary real numbers. x_(u)(m) is abase sequence for generating the sequence r(n). ‘u’ is a valuerepresenting a sequence index and is in a one-to-one mapping relationwith the combination of u₀, u₁, . . . , u_(k).

Here, u_(k) is a component for shifting the phase of the entiresequences and gives no effect in generating the sequences. Thus,Equation 4 may be expressed by the following form:r(n)=x _(u)(n), 0≦n≦N−1, x _(u)(m)=e ^(−j(u) ⁰ ^(m) ^(k) ^(+u) ¹ ^(m)^(k-1) ^(+ . . . +u) ^(k-1) ^(m) ¹ ⁾  [Equation 5]

In a difference example, a closed-form generating equation with respectto a sequence r(n) obtained by approximating or quantizing a phase valuein the sequence of Equation 4 can be expressed as follows:r(n)=x _(u)(n), 0≦n≦N−1, x _(u)(m)=e ^(−j*quan(u) ⁰ ^(m) ^(k) ^(+u) ¹^(m) ^(k-1) ^(+ . . . +u) ^(k-1) ^(m) ¹ ^(+u) ^(k) ⁾  [Equation 6]

where m=0, 1, . . . , N−1, ‘N’ denotes the length of the sequence r(n),and u₀, u₁, . . . , u_(k) denote arbitrary real numbers. quan(.) denotesa quantization function which means approximating or quantizing to aparticular value.

A real value and an imaginary value of the results of the sequence inEquation 6 may be approximated/quantized as shown below:

$\begin{matrix}{{{r(n)} = {x_{u}(n)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\frac{1}{\sqrt{p_{n}}}{{quan}\left( {\mathbb{e}}^{- {j{({{u_{0}m^{k}} + {u_{1}m^{k - 1}} + \;\ldots\; + {u_{k - 1}m^{1}} + u_{k}})}}} \right)}}}} & \left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N−1, and p_(n) denotes a normalization factor forregulating the amplitude of a generated sequence.

In Equation 6, values on a complex unit circle that a e^(−jθ) may haveare quantized to Nq number. The quantized values can be approximated tothe coordinates of QPSK {0.7071+j0.7071, 0.7071−j0.7071,−0.7071+j0.7071), −0.7071−j0.7071}, or approximated to {exp(−j*2*π*0/8), exp(−j*2*π*⅛), exp(−j*2*π* 2/8), exp(−j*2*π*⅜), exp(−j*2*π* 4/8),exp(−j*2*π*⅝), exp(−j*2*π* 6/8), exp(−j*2*π*⅞)} in the form of 8-PSK.

In this case, according to the approximating methods, the values can beapproximated to the closest values, to the same or the closest smallvalues, or to the same or the closest large values.

In Equation 7, a real value and an imaginary value generated from thevalue of exponential function are approximated to the closest particularconstellation. That is, for example, they are approximated to M-PSK orM-QAM. In addition, the real value and the imaginary value may beapproximated to {+1, −1, 0} through a sign function which outputs thesign of the value.

In Equations 6 and 7, in order to approximate to the closest QPSK, thevalue u_(k) may be set to be π*¼. In addition, a round functionsignifying rounding as a particular form of the quantization functionmay be used. The quantization function may be used at a phase portion ofan exponential function or at the entire exponential function.

Variables may be set according to a particular criterion to generatesequences from the generating equations. The criterion may consider CMor correlation characteristics. For example, a CM value and a thresholdof cross-correlation may be set to generate sequences.

A detailed generating equations for generating sequences from theabove-described general generating equations will now be described.

First Embodiment Simple Polynomial Expression Form (k=3)

The following generating equation may be selected:r(n)=x _(u)(n), 0≦n≦N−1, x _(u)(m)=e ^(−j(u) ⁰ ^(m) ³ ^(+u) ¹ ^(m) ²^(+u) ² ^(m) ¹ ⁾  [Equation 8]

where m=0, 1, . . . , N−1, ‘N’ denotes the length of the sequence r(n),and u₀, u₁, u₂ denote arbitrary real numbers.

Second Embodiment Modified ZC Sequence

The following generating equation may be selected:

$\begin{matrix}{{{r(n)} = {x_{u}(n)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi{({{u_{0}m^{k}} + {u_{1}m^{k - 1}} + \;\ldots\; + {u_{k - 1}m^{1}}})}}{N}}}} & \left\lbrack {{Equation}\mspace{14mu} 9} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N−1, ‘N’ denotes the length of the sequence r(n),and u₀, u₁, . . . , u_(k-1) denote arbitrary real numbers.

This generating equation has the following advantages. Firstly, ZCsequences having good characteristics that can be created with thelength N can be included in an available sequence set. For example, ifk=2, u₁=0 and u₀ is an integer, it is equivalent to ZC sequences when Nin Equation 1 is an even number. If k=2, u₁ and u₀ are integers, andu₁=u₀, it is equivalent to ZC sequences when N in Equation 1 is an oddnumber. Second, sequences having good characteristics as close as theEuclidean distance of original optimized ZC sequences.

Third Embodiment Cyclic Extended Corrected ZC Sequence

The following generating equation may be selected:

$\begin{matrix}{{{r(n)} = {x_{u}\left( {n\;{mod}\; N_{ZC}} \right)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi{({{u_{0}m^{k}} + {u_{1}m^{k - 1}} + \;\ldots\; + {u_{k - 1}m^{1}}})}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 10} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N−1, N denotes the length of the sequence r(n),and u₀, u₁, . . . , u_(k-1) denote arbitrary real numbers. N_(ZC) is thelargest prime number among natural numbers smaller than N. Thisgenerating equation is advantageous in that an existing ZC sequence canbe included in an available sequence set. For example, if k=2, u₁ and u₀are integers, and u₁=u₀, it is equivalent to a value obtained by cyclicextending the ZC sequence.

Fourth Embodiment Truncated Modified ZC Sequence

The following generating equation may be selected:

$\begin{matrix}{{{r(n)} = {x_{u}(n)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi{({{u_{0}m^{k}} + {u_{1}m^{k - 1}} + \;\ldots\; + {u_{k - 1}m^{1}}})}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 11} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N−1, N denotes the length of the sequence r(n),and u₀, u₁, . . . , u_(k-1) denote arbitrary real numbers. N_(ZC) is thelargest prime number among natural numbers larger than N. Thisgenerating equation is advantageous in that an existing ZC sequence canbe included in an available sequence set. For example, if k=2, and u₁and u₀ are integers, it is equivalent to a value obtained by truncatingthe ZC sequence.

Fifth Embodiment Modified ZC Sequence Having a Restriction

The following generating equation may be selected:

$\begin{matrix}{{{r(n)} = {x_{u}(n)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{u_{0}m^{k}} + {u_{1}m^{k - 1}} + \;\ldots\; + {u_{k - 1}m^{1}}})}}}{N}}}} & \left\lbrack {{Equation}\mspace{14mu} 12} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N−1, N denotes the length of the sequence r(n),u₀, u₁, . . . , u_(k-1) denote arbitrary integers, and ‘a’ denotes anarbitrary real number. ‘a’ serves to restrict granularity of thevariables u₀, u₁, . . . , u_(k-1). Because the granularity of thevariables u₀, u₁, . . . , u_(k-1) can be changed into the unit ofinteger through such restriction, a memory required for storing sequenceinformation can be reduced.

Sixth Embodiment Modified ZC Sequence Having Two Restrictions

The following generating equation may be selected:

$\begin{matrix}{{{r(n)} = {x_{u}(n)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{b_{0}u_{0}m^{k}} + {b_{1}u_{1}m^{k - 1}} + \;\ldots\; + {b_{k - 1}u_{k - 1}m^{1}}})}}}{N}}}} & \left\lbrack {{Equation}\mspace{14mu} 13} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N−1, N denotes the length of the sequence r(n),u₀, u₁, . . . , u_(k-1) denote arbitrary integers, ‘a’ denotes anarbitrary real number, and b₀, b₁, . . . , b_(k-1) denote arbitrary realnumbers. ‘a’ serves to restrict granularity of the variables u₀, u₁, . .. , u_(k-1). It may differently restrict the variables through b₀, b₁, .. . , b_(k-1). A memory required for storing sequence information can bereduced by changing the granularity of the variables u₀, u₁, . . . ,u_(k-1) into the unit of integer through the two restrictions, and asequence of better characteristics can be obtained by adjusting thegranularity by variables.

Seventh Embodiment Modified ZC Sequence (k=3) Having Two Restrictions

The following creation formula can be selected:

$\begin{matrix}{{{r(n)} = {x_{u}(n)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{b_{0}u_{0}m^{3}} + {b_{1}u_{1}m^{2}} + {b_{2}u_{2}m^{1}}})}}}{N}}}} & \left\lbrack {{Equation}\mspace{14mu} 14} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N−1, N denotes the length of the sequence r(n),u₀, u₁, u₂ denote arbitrary integers, ‘a’ denotes an arbitrary realnumber, and b₀, b₁, b₂ denote arbitrary integers. ‘a’ serves to restrictgranularity of the variables u₀, u₁, u₂. It may differently restrict thevariables through b₀, b₁, b₂.

Eighth Embodiment Modified ZC Sequence Having One Restriction and CyclicExtension

The following generating equation may be selected:

$\begin{matrix}{{{r(n)} = {x_{u}\left( {n\;{mod}\; N_{ZC}} \right)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{u_{0}m^{k}} + {u_{1}m^{k - 1}} + \;\ldots\; + {u_{k - 1}m^{1}}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 15} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N_(ZC)−1, N denotes the length of the sequencer(n), u₀, u₁, . . . , u_(k-1) denote arbitrary integers, ‘a’ denotes anarbitrary real number, and N_(ZC) denotes the largest prime number amongnatural numbers smaller than ‘N’. ‘a’ serves to restrict granularity ofthe variables u₀, u₁, . . . , u_(k-1). Because the granularity of thevariables u₀, u₁, . . . , u_(k-1) can be changed into the unit ofinteger through such restriction, a memory required for storing sequenceinformation can be reduced.

Ninth Embodiment Modified ZC Sequence Having Two Restrictions and CyclicExtension

The following generating equation may be selected:

$\begin{matrix}{{{r(n)} = {x_{u}\left( {n\;{mod}\; N_{ZC}} \right)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{b_{0}u_{0}m^{k}} + {b_{1}u_{1}m^{k - 1}} + \;\ldots\; + {b_{k - 1}u_{k - 1}m^{1}}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 16} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N_(ZC)−1, N denotes the length of the sequencer(n), u₀, u₁, . . . , u_(k-1) denote arbitrary integers, ‘a’ denotes anarbitrary real number, b₀, b₁, . . . , b_(k-1) denote arbitraryintegers, and N_(ZC) denotes the largest prime number among naturalnumbers smaller than ‘N’. ‘a’ serves to restrict granularity of thevariables u₀, u₁, . . . , u_(k-1). It may differently restrict thevariables through b₀, b₁, . . . , b_(k-1). A memory required for storingsequence information can be reduced by changing the granularity of thevariables u₀, u₁, . . . , u_(k-1) into the unit of integer through thetwo restrictions, and a sequence of better characteristics can beobtained by adjusting the granularity by variables.

10th Embodiment Modified ZC Sequence Having Two Restrictions (k=3) andCyclic Extension

The following generating equation may be selected:

$\begin{matrix}{{{r(n)} = {x_{u}\left( {n\;{mod}\; N_{ZC}} \right)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{b_{0}u_{0}m^{3}} + {b_{1}u_{1}m^{2}} + \;{b_{2}u_{2}m^{1}}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 17} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N−1, N denotes the length of the sequence r(n),u₀, u₁, u₂ denote arbitrary integers, ‘a’ denotes an arbitrary realnumber, b₀, b₁, b₂ denote arbitrary integers, and N_(ZC) denotes thelargest prime number among natural numbers smaller than N. ‘a’ serves torestrict granularity of the variables u₀, u₁, u₂. It may differentlyrestrict the variables through b₀, b₁, b₂.

11th Embodiment Modified ZC Sequence Having One Restriction andTruncation

The following generating equation may be selected:

$\begin{matrix}{{{r(n)} = {x_{u}(n)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{u_{0}m^{k}} + {u_{1}m^{k - 1}} + \;\ldots\; + {u_{k - 1}m^{1}}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 18} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N_(ZC)−1, N is the length of the sequence r(n),u₀, u₁, . . . , u_(k-1) are arbitrary integers, ‘a’ is an arbitrary realnumber, and N_(ZC) is the largest prime number among natural numberslarger than N. ‘a’ serves to restrict granularity of the variables u₀,u₁, . . . , u_(k-1). Because the granularity of the variables u₀, u₁, .. . , u_(k-1) can be changed into the unit of integer through suchrestriction, a memory required for storing sequence information can bereduced.

12th Embodiment Modified ZC Sequence Having Two Restrictions andTruncation

The following generating equation may be selected.

$\begin{matrix}{{{r(n)} = {x_{u}(n)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{b_{0}u_{0}m^{k}} + {b_{1}u_{1}m^{k - 1}} + \;\ldots\; + {b_{k - 1}u_{k - 1}m^{1}}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 19} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N_(ZC)−1, N is the length of the sequence r(n),u₀, u₁, . . . , u_(k-1) are arbitrary integers, ‘a’ is an arbitrary realnumber, b₀, b₁, . . . , b_(k-1) are arbitrary integers, and N_(ZC) isthe smallest prime number among natural numbers larger than N. ‘a’serves to restrict granularity of the variables u₀, u₁, . . . , u_(k-1).It may differently restrict the variables through b₀, b₁, . . . ,b_(k-1). A memory required for storing sequence information can bereduced by changing the granularity of the variables u₀, u₁, . . . ,u_(k-1) into the unit of integer through the two restrictions, and asequence of better characteristics can be obtained by adjusting thegranularity by variables.

13th Embodiment Modified ZC Sequence Having Two Restrictions (k=3) andTruncation

The following generating equation may be selected:

$\begin{matrix}{{{r(n)} = {x_{u}(n)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{b_{0}u_{0}m^{3}} + {b_{1}u_{1}m^{2}} + \;{b_{2}u_{2}m^{1}}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 20} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N_(ZC)−1, N is the length of the sequence r(n),u₀, u₁, u₂ are arbitrary integers, ‘a’ is an arbitrary real number, b₀,b₁, b₂ are arbitrary integers, and N_(ZC) is the smallest prime numberamong natural numbers larger than N. ‘a’ serves to restrict granularityof the variables u₀, u₁, u₂. It may differently restrict the variablesthrough b₀, b₁, b₂.

14th Embodiment Cyclic Extension in Consideration of Cyclic Shift inTime Domain

In an OFDMA system or SC-FDMA system, the number of available sequencescan be increased by using cyclic shifts for each root sequence. Besidesthe cyclic shift, a start point for generating a sequence can becombined with a particular frequency index so as to be defined. This isa restriction of forcibly adjusting start points overlapped by differentsequences in the frequency domain, having an advantage in that thecorrelation characteristics of the modified ZC sequence having one ormore restrictions can be supported as it is.

For example, the following sequence generating equation may be selected:

$\begin{matrix}{{{r(n)} = {{\mathbb{e}}^{j\;\alpha\; n}{x_{u}\left( {\left( {n + \theta} \right){mod}\; N_{ZC}} \right)}}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{\alpha{({{u_{0}m^{k}} + {u_{1}m^{k - 1}} + \;\ldots\; + {u_{k - 1}m^{1}}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 21} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N_(ZC)−1, N is the length of the sequence r(n),u₀, u₁, . . . , u_(k-1) are arbitrary integers, ‘a’ is an arbitrary realnumber, and N_(ZC) is the largest prime number among natural numberssmaller than N. e^(jan) is an expression, in the frequency domain, ofperforming cyclic shift by ‘α’ at the time domain. θ is a shift offsetvalue and indicates performing of cyclic extension after shifting by θ.If Equation 21 is expressed in the frequency domain, θ indicates a shiftvalue of a frequency index.

For another example, the following sequence generating equation may beselected:

$\begin{matrix}{{{r(n)} = {{\mathbb{e}}^{{j\alpha}\; n}{x_{u}\left( {\left( {n + \theta} \right){mod}\; N_{ZC}} \right)}}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{b_{0}u_{0}m^{k}} + {b_{1}u_{1}m^{k - 1}} + \ldots + {b_{k - 1}u_{k - 1}m^{1}}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 22} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N_(ZC)−1, N is the length of the sequence r(n),u₀, u₁, . . . , u_(k-1) are arbitrary integers, ‘a’ is an arbitrary realnumber, b₀, b₁, . . . , b_(k-1) are arbitrary integers, and N_(ZC) isthe largest prime number among natural numbers smaller than N. e^(jan)is an expression, in the frequency domain, of performing cyclic shift by‘α’ at the time domain. θ is a shift offset value and indicatesperforming of cyclic extension after shifting by θ.

For a still another example, the following sequence generating equationmay be selected:

$\begin{matrix}{{{r(n)} = {{\mathbb{e}}^{{j\alpha}\; n}{x_{u}\left( {\left( {n + \theta} \right){mod}\; N_{ZC}} \right)}}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{b_{1}u_{0}m^{3}} + {b_{1}u_{1}m^{2}} + \ldots + {b_{2}u_{2}m^{1}}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 23} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N_(ZC)−1, N is the length of the sequence r(n),u₀, u₁, u₂ are arbitrary integers, ‘a’ is an arbitrary real number, b₀,b₁, b₂ are arbitrary integers, and N_(ZC) is the largest prime numberamong natural numbers smaller than N. e^(jan) is an expression, in thefrequency domain, of performing cyclic shift by ‘α’ at the time domain.θ is a shift offset value.

15th Embodiment Truncation in Consideration of Cyclic Shift in TimeDomain

For example, the following sequence generating equation may be selected:

$\begin{matrix}{{{r(n)} = {{\mathbb{e}}^{j\;\alpha\; n}{x_{u}(n)}}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{u_{0}m^{k}} + {u_{1}m^{k - 1}} + \ldots + {u_{k - 1}m^{1}}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 24} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N_(ZC)−1, N is the length of the sequence r(n),u₀, u₁, . . . , u_(k-1) are arbitrary integers, ‘a’ is an arbitrary realnumber, and N_(ZC) is the largest prime number among natural numberssmaller than N. e^(jan) is an expression, in the frequency domain, ofperforming cyclic shift by ‘α’ at the time domain.

For another example, the following sequence generating equation may beselected.

$\begin{matrix}{{{r(n)} = {{\mathbb{e}}^{{j\alpha}\; n}{x_{u}(n)}}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = e^{{- j}\frac{\pi\;{a{({{b_{0}u_{0}m^{k}} + {b_{1}u_{1}m^{k}} + \ldots + {b_{k - 1}u_{k - 1}m^{1}}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 25} \right\rbrack\end{matrix}$

For a still another example, the following sequence generating equationmay be selected.

$\begin{matrix}{{{r(n)} = {{\mathbb{e}}^{{j\alpha}\; n}{x_{u}(n)}}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{b_{1}u_{0}m^{3}} + {b_{1}u_{1}m^{2}} + {b_{2}u_{2}m^{1}}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 26} \right\rbrack\end{matrix}$

In Equation 26, if k=3, a=0.125, b₀=2, and b₁=b₂=1=1, then the followingequation can be obtained.

$\begin{matrix}{{{r(n)} = {{\mathbb{e}}^{{j\alpha}\; n}{x_{u}(n)}}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{{\pi 0}{.125}{({{2u_{0}m^{3}} + {u_{1}m^{2}} + {u_{2}m^{1}}})}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 27} \right\rbrack\end{matrix}$

IV. Generation of Sequence

In order to show an example of generating a sequence, the followingsequence generating equation is considered:

$\begin{matrix}{{{r(n)} = {x_{u}\left( {\left( {n + \theta} \right){mod}\; N_{ZC}} \right)}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{{\pi 0}{.125}{({{2u_{0}m^{3}} + {u_{1}m^{2}} + {u_{2}m^{1}}})}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 28} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N_(ZC)−1, N is the length of the sequence r(n),u₀, u₁, u₂ are arbitrary integers, θ is a shift offset value, and N_(ZC)is the largest prime number among natural numbers smaller than N. ThisEquation is obtained by defining α=0, k=3, a=0.125, b₀=2, b₁=b₂=1. Thereason of selecting a=0.125 is to reduce the amount of calculation.Namely, because 0.125 is ⅛, it can be implemented by three times of bitshifting operation,

The variables u₀, u₁, u₂ are determined by using a CM and a thresholdvalue of cross-correlation.

First, generation of a sequence with a length of N=12 will now bedescribed.

When a CM reference was set as 1.2 dB and the threshold ofcross-correlation was set as 0.6, the values of the variables u₀, u₁, u₂and CMs of corresponding sequences obtained from the generating equationare as shown in below table.

TABLE 2 Sequence Index u₀ u₁ u₂ CM [dB] 0 0 9 8 0.17 1 0 32 32 0.85 2 040 40 0.43 3 0 48 48 0.43 4 0 56 56 0.85 5 0 80 80 0.17 6 0 19 10 1.08 70 26 0 1.12 8 0 61 0 0.87 9 0 68 3 1.18 10 1 78 22 1.11 11 2 25 60 0.9912 3 62 2 1.15 13 3 73 4 1.15 14 3 80 37 1.10 15 4 82 8 1.18 16 11 38 861.18 17 12 65 75 1.12 18 14 73 52 1.20 19 16 83 61 1.05 20 18 34 11 1.1121 18 50 41 1.16 22 22 17 44 0.88 23 25 61 36 1.14 24 25 88 11 1.17 2527 39 5 1.12 26 32 23 85 1.12 27 34 17 52 1.10 28 38 36 31 1.04 29 40 68 1.18

In the above table, sequences of the index 0 to 5 refer to a set ofsequences satisfying the CM reference, among ZC sequences generated byapplying the conventional cyclic extension.

Table 3 shows real number values of sequences generated from Table 2,and Table 4 shows imaginary number values of sequences generated fromTable 2.

TABLE 3 Sequence index n 0 1 0.84125 −0.14231 −0.95949 0.84125 −0.654860.84125 −0.95949 −0.14231 0.84125 1 1 1 1 −0.65486 0.84125 0.41542−0.65486 −0.95949 −0.65486 0.41542 0.84125 −0.65486 1 1 2 1 −0.95949−0.65486 −0.14231 −0.95949 0.41542 −0.95949 −0.14231 −0.65486 −0.95949 11 3 1 −0.95949 −0.65486 −0.14231 −0.95949 0.41542 −0.95949 −0.14231−0.65486 −0.95949 1 1 4 1 −0.65486 0.84125 0.41542 −0.65486 −0.95949−0.65486 0.41542 0.84125 −0.65486 1 1 5 1 0.84125 −0.14231 −0.959490.84125 −0.65486 0.84125 −0.95949 −0.14231 0.84125 1 1 6 1 0.51027−0.95949 0.62747 0.95949 0.99427 0.14231 −0.38268 −0.65486 −0.03569−0.65486 1 7 1 0.59928 −0.84125 −0.47925 −0.65486 −0.34946 −0.415420.071339 −0.95949 0.97715 0.14231 1 8 1 −0.57032 −0.75575 0.73189−0.95949 −0.51027 −0.98982 0.99427 0.41542 0.89423 −0.54064 1 9 1−0.82142 −0.87768 −0.98411 0 −0.03569 0.99745 0.94883 0.65486 −0.948830.34946 1 10 1 −0.87768 0.75575 −0.47925 −0.41542 0.70711 0.54064−0.99745 0.41542 −0.34946 0.90963 1 11 1 −0.99936 −0.90963 0.860010.84125 0.82142 0.90963 0.62747 −0.95949 0.62747 −0.54064 1 12 1−0.80054 −0.28173 0.70711 0.65486 0.70711 0.75575 0.97715 0.84125−0.99745 −0.90963 1 13 1 −0.98411 0.98982 −0.17755 0.84125 −0.03569−0.90963 0.44762 0.41542 −0.57032 0.28173 1 14 1 −0.3158 −0.99745−0.62747 −0.28173 −0.44762 0.99745 −0.92388 0.14231 −0.92388 −0.59928 115 1 −0.93695 −0.41542 −0.93695 −0.95949 −0.93695 0.65486 −0.07134−0.95949 −0.21257 −1 1 16 1 0.47925 0.54064 0.21257 −0.84125 −0.97715−0.28173 0.70711 −0.14231 0.99745 −0.98982 1 17 1 0.90963 −0.87768−0.21257 −0.54064 −0.84125 −0.34946 0.59928 0.65486 0.54064 −0.93695 118 1 0.68142 −0.98982 0.86001 1 −0.1069 0.90963 0.96894 −0.65486 0.1069−0.98982 1 19 1 1 0.97715 0.34946 −0.90963 0 −0.47925 0.80054 0.959490.65486 −0.93695 1 20 1 −0.96894 −0.97715 −0.94883 −0.90963 −0.24731−0.99745 0.92388 −0.84125 −0.44762 −0.99745 1 21 1 −0.17755 0.0713390.17755 −0.90963 0.51027 0.34946 −0.24731 0.14231 0.17755 −0.21257 1 221 −0.82142 0.75575 −0.68142 −0.95949 0.86001 0.98982 0.1069 0.41542−0.44762 0.54064 1 23 1 0.51027 0.90963 −0.82142 −0.95949 0.3158 0.909630.62747 −0.14231 0.99427 0.28173 1 24 1 0.57032 −0.80054 −0.62747−0.90963 −0.44762 −0.07134 −0.73189 0.95949 0.51027 −0.21257 1 25 1−0.93695 −0.80054 −0.65486 −0.28173 0.97715 −0.87768 −0.28173 −1 0.70711−0.07134 1 26 1 0.98982 −0.80054 −0.93695 −0.28173 0.65486 0.59928−0.99745 −0.84125 0.98982 −0.97715 1 27 1 0.17755 0.90963 0.38268−0.95949 0.38268 −0.28173 −0.44762 −0.65486 0.62747 0.98982 1 28 10.38268 −0.70711 0.98411 −0.75575 0.98411 −0.34946 −0.68142 −0.841250.94883 0.97715 1 29 1 −0.97715 0.65486 −0.21257 0.41542 0.80054−0.41542 0.80054 −0.14231 −0.97715 −0.84125 1

TABLE 4 Sequence n index 0 1 2 3 4 5 6 7 8 9 10 11 0 0 −0.54064 −0.989820.28173 0.54064 −0.75575 0.54064 0.28173 −0.98982 −0.54064 0 0 1 0−0.75575 −0.54064 −0.90963 0.75575 −0.28173 0.75575 −0.90963 −0.54064−0.75575 0 0 2 0 −0.28173 −0.75575 0.98982 0.28173 0.90963 0.281730.98982 −0.75575 −0.28173 0 0 3 0 0.28173 0.75575 −0.98982 −0.28173−0.90963 −0.28173 −0.98982 0.75575 0.28173 0 0 4 0 0.75575 0.540640.90963 −0.75575 0.28173 −0.75575 0.90963 0.54064 0.75575 0 0 5 00.54064 0.98982 −0.28173 −0.54064 0.75575 −0.54064 −0.28173 0.989820.54064 0 0 6 0 −0.86001 0.28173 −0.77864 0.28173 0.1069 −0.989820.92388 −0.75575 −0.99936 −0.75575 0 7 0 −0.80054 0.54064 −0.87768−0.75575 0.93695 −0.90963 −0.99745 −0.28173 0.21257 0.98982 0 8 0−0.82142 −0.65486 −0.68142 0.28173 0.86001 −0.14231 0.1069 −0.90963−0.44762 0.84125 0 9 0 −0.57032 0.47925 0.17755 −1 0.99936 −0.07134−0.3158 0.75575 −0.3158 0.93695 0 10 0 0.47925 −0.65486 0.87768 −0.90963−0.70711 −0.84125 −0.07134 −0.90963 −0.93695 0.41542 0 11 0 0.035692−0.41542 0.51027 −0.54064 −0.57032 −0.41542 −0.77864 −0.28173 −0.77864−0.84125 0 12 0 −0.59928 0.95949 −0.70711 0.75575 −0.70711 −0.65486−0.21257 −0.54064 −0.07134 −0.41542 0 13 0 −0.17755 0.14231 0.984110.54064 0.99936 −0.41542 −0.89423 −0.90963 0.82142 0.95949 0 14 00.94883 0.071339 0.77864 −0.95949 0.89423 0.071339 −0.38268 −0.989820.38268 0.80054 0 15 0 0.34946 −0.90963 0.34946 0.28173 0.34946 0.755750.99745 −0.28173 −0.97715 0 0 16 0 0.87768 0.84125 0.97715 −0.54064−0.21257 0.95949 0.70711 0.98982 −0.07134 −0.14231 0 17 0 0.41542−0.47925 −0.97715 −0.84125 −0.54064 −0.93695 0.80054 0.75575 −0.841250.34946 0 18 0 0.73189 0.14231 0.51027 0 0.99427 −0.41542 0.24731−0.75575 −0.99427 0.14231 0 19 0 0 −0.21257 −0.93695 0.41542 −1 −0.877680.59928 −0.28173 0.75575 −0.34946 0 20 0 −0.24731 0.21257 −0.3158−0.41542 0.96894 0.071339 −0.38268 0.54064 −0.89423 −0.07134 0 21 00.98411 −0.99745 0.98411 0.41542 0.86001 0.93695 0.96894 0.98982−0.98411 −0.97715 0 22 0 0.57032 0.65486 −0.73189 0.28173 0.510270.14231 −0.99427 −0.90963 −0.89423 −0.84125 0 23 0 0.86001 −0.41542−0.57032 0.28173 −0.94883 −0.41542 0.77864 −0.98982 −0.1069 0.95949 0 240 0.82142 −0.59928 −0.77864 −0.41542 −0.89423 0.99745 −0.68142 0.28173−0.86001 0.97715 0 25 0 0.34946 −0.59928 −0.75575 −0.95949 −0.21257−0.47925 −0.95949 0 0.70711 −0.99745 0 26 0 0.14231 −0.59928 −0.34946−0.95949 −0.75575 −0.80054 0.071339 −0.54064 −0.14231 0.21257 0 27 00.98411 −0.41542 −0.92388 −0.28173 −0.92388 0.95949 −0.89423 −0.75575−0.77864 0.14231 0 28 0 0.92388 0.70711 −0.17755 0.65486 0.17755 0.93695−0.73189 0.54064 0.3158 −0.21257 0 29 0 0.21257 0.75575 0.97715 0.909630.59928 0.90963 0.59928 −0.98982 0.21257 −0.54064 0

If N=12 and when sequences generated by the proposed generating equationand the ZC sequences generated by applying the conventional cyclicextension, six sequences satisfying QPSK CM criteria 1.2 dB areincluded.

Table 5 shows a comparison between the ZC sequence generated by applyingthe conventional cyclic extension and the proposed sequences.

TABLE 5 Conventional ZC Proposed sequence Sequence Num. of Sequences 1030 Num. of Sequences < CM 1.2 dB 6 30 Max. CM [dB] 1.50 1.20 Max. Cross.Cor. 0.44 0.60 Mean Cross. Cor. 0.25 0.25 Median Cross. Cor. 0.28 0.24

It is noted that, when the sequences are generated by the proposedmethod, the number of available sequences can be increased while thecross-correlation characteristics are substantially the same. Whenfrequency hopping in an actual environment is considered, a block errorrate (BLER) performance becomes better as a mean correlation value islower. Because mean correlations of both sequences are the same, theBLER performance is the same.

Generation of a sequence with a length N=24 will now be described.

The below table shows variables u₀, u₁, u₂ obtained from the generatingequation and corresponding CMs when the CM reference is set to be 1.2 dBand the threshold value of the cross-correlation is set to be 0.39.

TABLE 6 Sequence Index u₀ u₁ u₂ CM [dB] 0 0 8 8 −0.09 1 0 32 32 0.83 2 048 48 0.68 3 0 64 64 0.38 4 0 72 72 0.49 5 0 88 88 0.18 6 0 96 96 0.18 70 112 112 0.49 8 0 120 120 0.38 9 0 136 136 0.68 10 0 152 152 0.83 11 0176 176 −0.09 12 0 6 17 1.11 13 0 6 182 0.87 14 0 25 16 1.14 15 0 29 820.95 16 0 35 132 0.92 17 0 44 27 0.83 18 0 48 4 1.01 19 0 54 18 1.13 200 54 122 1.14 21 0 58 0 1.07 22 0 64 14 0.61 23 0 68 21 0.98 24 0 88 110.58 25 0 96 116 0.63 26 0 112 0 0.49 27 0 126 133 1.05 28 0 130 15 1.0729 0 178 39 1.11

In the above table, sequences of the sequence indexes 0 to 11 refer to aset of sequences satisfying the CM criteria among the ZC sequencesgenerated by applying the conventional cyclic extension.

Table 7 shows real number values of the sequences generated from Table6, and Table 8 shows imaginary values of the generated sequences.

TABLE 7 Sequence n index 0 1 2 3 4 5 6 7 8 0 1 0.96292 0.68255 −0.06824−0.91721 −0.57668 0.85442 0.20346 −0.91721 1 1 0.46007 −0.99069 0.96292−0.06824 −0.77571 −0.57668 0.68255 −0.06824 2 1 −0.06824 0.20346−0.91721 −0.77571 0.85442 −0.99069 −0.33488 −0.77571 3 1 −0.576680.96292 0.85442 −0.99069 0.20346 −0.33488 −0.06824 −0.99069 4 1 −0.775710.46007 −0.57668 0.85442 0.68255 0.20346 0.96292 0.85442 5 1 −0.99069−0.91721 0.68255 0.20346 0.46007 0.96292 −0.77571 0.20346 6 1 −0.99069−0.91721 0.68255 0.20346 0.46007 0.96292 −0.77571 0.20346 7 1 −0.775710.46007 −0.57668 0.85442 0.68255 0.20346 0.96292 0.85442 8 1 −0.576680.96292 0.85442 −0.99069 0.20346 −0.33488 −0.06824 −0.99069 9 1 −0.068240.20346 −0.91721 −0.77571 0.85442 −0.99069 −0.33488 −0.77571 10 10.46007 −0.99069 0.96292 −0.06824 −0.77571 −0.57668 0.68255 −0.06824 111 0.96292 0.68255 −0.06824 −0.91721 −0.57668 0.85442 0.20346 −0.91721 121 0.92388 0.54845 −0.22014 −0.94226 −0.64424 0.6572 0.71908 −0.85442 131 −0.99767 0.94226 −0.68255 0.068242 0.73084 −0.94226 −0.06824 1 14 10.76482 −0.63109 −0.0512 −0.06824 0.86317 −0.2698 −0.66998 −0.33488 15 1−0.31874 0.068242 −0.71908 0.57668 0.86317 0.46007 −0.8799 0.46007 16 1−0.95817 0.81697 0.91028 0.96292 0.47516 −0.88789 0.47516 0.96292 17 10.35092 −0.70711 −0.2862 0.2698 −0.61775 −0.03414 −0.69493 0.068242 18 10.63109 −0.96292 0.2698 0.68255 −0.3984 0.068242 −0.97908 −0.91721 19 10.33488 −0.3984 −0.97908 −0.96292 0.85442 −0.88789 −0.97908 0.20346 20 1−0.99069 −5.8E−16 −0.3984 −0.46007 −0.46007 −0.13617 −0.99767 0.96292 211 0.54845 −0.68255 −0.87166 −0.99069 0.93028 −0.46007 −0.16991 0.8544222 1 0.23676 0.13617 −0.42948 0.91721 −0.97157 −0.99767 0.23676 −0.9172123 1 0.051199 0.60424 0.50492 0.3984 0.8267 0.99942 −0.95817 −0.20346 241 −0.11923 0.99476 0.051199 0.94226 0.69493 0.23676 0.89561 −0.96292 251 −0.88789 −0.46007 −0.2698 −0.91721 0.81697 −0.20646 0.99767 0.20346 261 −0.33488 0.20346 −0.06824 0.68255 −0.77571 0.96292 0.85442 −0.99069 271 −0.2862 0.83618 0.50492 0.88789 −0.66998 −0.99942 −0.35092 0.33488 281 −0.78637 −0.99942 −0.31874 0.3984 0.97547 0.97157 −0.8267 0.91721 29 1−0.84542 0.60424 −0.47516 0.51958 −0.71908 0.95314 −0.93641 0.33488Sequence n index 9 10 11 12 13 14 15 16 0 0.96292 −0.77571 0.68255−0.77571 0.96292 −0.91721 0.20346 0.85442 1 0.46007 −0.91721 −0.99069−0.91721 0.46007 −0.06824 0.68255 −0.57668 2 −0.6824 −0.57668 0.20346−0.57668 −0.06824 −0.77571 −0.33488 −0.99069 3 −0.57668 0.68255 0.962920.68255 −0.57668 −0.99069 −0.06824 −0.33488 4 −0.77571 −0.99069 0.46007−0.99069 −0.77571 0.85442 0.96292 0.20346 5 −0.99069 −0.33488 −0.91721−0.33488 −0.99069 0.20346 −0.77571 0.96292 6 −0.99069 −0.33488 −0.91721−0.33488 −0.99069 0.20346 −0.77571 0.96292 7 −0.77571 −0.99069 0.46007−0.99069 −0.77571 0.85442 0.96292 0.20346 8 −0.57668 0.68255 0.962920.68255 −0.57668 −0.99069 −0.06824 −0.33488 9 −0.06824 −0.57668 0.20346−0.57668 −0.06824 −0.77571 −0.33488 −0.99069 10 0.46007 −0.91721−0.99069 −0.91721 0.46007 −0.06824 0.68255 −0.57668 11 0.96292 −0.775710.68255 −0.77571 0.96292 −0.91721 0.20346 0.85442 12 −0.08527 0.83618−0.99287 0.81697 −0.61775 0.54845 −0.64424 0.85442 13 0.13617 −0.88789−0.85442 −0.20346 0.3984 0.73084 0.85442 0.85442 14 0.78637 0.13617−0.31874 −0.33488 0.95817 0.88789 0.92388 0.85442 15 −0.76482 0.775710.99636 0.99069 0.22014 −0.91721 0.89561 −0.06824 16 0.91028 0.81697−0.95817 1 −0.08527 −0.51958 0.18671 0.85442 17 −0.56265 −0.36685−0.15306 0.81697 0.5341 −0.97157 0.99985 0.20346 18 −0.51958 0.576680.81697 0.85442 0.3984 −0.20346 −0.99767 −0.91721 19 −0.46007 0.51958−0.2698 −0.20346 −0.91721 −0.94226 −1.7E−14 −0.57668 20 0.68255 0.99767−0.81697 0.77571 0.77571 −0.81697 0.99767 0.68255 21 0.10226 0.0682420.90307 −0.33488 −0.6572 0.77571 −0.97157 −0.57668 22 −0.90307 0.13617−0.97157 −1 0.75371 −0.73084 −0.30251 0.68255 23 −0.99287 0.953140.99636 −0.2698 0.98241 0.99476 −0.91028 0.20346 24 −0.64424 0.23676−0.08527 0.2698 0.31874 −0.23676 −0.01707 −0.33488 25 0.97908 0.0682420.97908 −0.57668 0.3984 −0.96292 −0.88789 0.46007 26 −0.57668 −0.917210.46007 0.46007 −0.91721 −0.57668 −0.99069 0.85442 27 0.99636 0.60424−0.8267 −0.63109 −0.92388 0.49011 −0.96738 −0.91721 28 0.99287 −0.102260.35092 −0.63109 0.11923 0.36685 0.8267 0.85442 29 0.66998 −0.90307−0.35092 0.88789 0.71908 −0.23676 −0.8799 −0.99069 Sequence n index 1718 19 20 21 22 23 0 −0.57668 −0.91721 −0.06824 0.68255 0.96292 1 1 1−0.77571 −0.06824 0.96292 −0.99069 0.46007 1 1 2 0.85442 −0.77571−0.91721 0.20346 −0.06824 1 1 3 0.20346 −0.99069 0.85442 0.96292−0.57668 1 1 4 0.68255 0.85442 −0.57668 0.46007 −0.77571 1 1 5 0.460070.20346 0.68255 −0.91721 −0.99069 1 1 6 0.46007 0.20346 0.68255 −0.91721−0.99069 1 1 7 0.68255 0.85442 −0.57668 0.46007 −0.77571 1 1 8 0.20346−0.99069 0.85442 0.96292 −0.57668 1 1 9 0.85442 −0.77571 −0.917210.20346 −0.06824 1 1 10 −0.77571 −0.06824 0.96292 −0.99069 0.46007 1 111 −0.57668 −0.91721 −0.06824 0.68255 0.96292 1 1 12 −0.99985 0.753710.085266 −0.94226 0.5341 0.83618 1 13 0.73084 0.3984 −0.20346 −0.85442−0.88789 0.13617 1 14 −0.69493 0.2698 −0.59054 0.96292 0.69493 0.51958 115 −0.92388 −0.96292 −0.414 0.99069 −0.91028 0.96292 1 16 −0.86317−0.13617 −0.59054 0.20346 −0.98822 0.88789 1 17 0.31874 0.93028 −0.93641−0.2698 −0.11923 −0.99476 1 18 0.73084 −0.96292 −0.2698 −0.77571 5.4E−15−0.68255 1 19 0.068242 −0.88789 0.81697 −0.46007 −0.06824 0.81697 1 200.96292 −0.99767 −0.13617 −0.46007 −0.46007 −0.3984 1 21 −0.953140.91721 0.79681 0.96292 −0.99942 −0.20346 1 22 0.83618 0.97908 −0.99942−0.46007 −0.99942 0.99767 1 23 −0.69493 0.79681 0.25331 0.94226 −0.38268−0.36685 1 24 −0.74238 0.99476 0.78637 8.57E−14 0.86317 −0.79681 1 252.11E−14 0.33488 0.51958 −0.57668 −0.51958 0.33488 1 26 0.96292 −0.775710.68255 −0.06824 0.20346 −0.33488 1 27 0.8267 −0.93028 −0.98241 0.3984−0.86317 −0.49011 1 28 0.22014 0.36685 −0.31874 0.73084 −0.61775 0.707111 29 −0.84542 −0.70711 −0.69493 −0.81697 −0.97547 −0.93028 1

TABLE 8 Sequence n index 0 1 2 3 4 5 6 7 8 0 0 −0.2698 −0.73084 −0.99767−0.3984 0.81697 0.51958 −0.97908 0.3984 1 0 −0.88789 0.13617 −0.26980.99767 0.63109 0.81697 0.73084 −0.99767 2 0 −0.99767 0.97908 0.39840.63109 0.51958 −0.13617 −0.94226 −0.63109 3 0 −0.81697 −0.2698 −0.51958−0.13617 −0.97908 −0.94226 0.99767 0.13617 4 0 −0.63109 −0.88789−0.81697 0.51958 0.73084 −0.97908 0.2698 −0.51958 5 0 −0.13617 −0.39840.73084 0.97908 −0.88789 −0.2698 −0.63109 −0.97908 6 0 0.13617 0.3984−0.73084 −0.97908 0.88789 0.2698 0.63109 0.97908 7 0 0.63109 0.887890.81697 −0.51958 −0.73084 0.97908 −0.2698 0.51958 8 0 0.81697 0.26980.51958 0.13617 0.97908 0.94226 −0.99767 −0.13617 9 0 0.99767 −0.97908−0.3984 −0.63109 −0.51958 0.13617 0.94226 0.63109 10 0 0.88789 −0.136170.2698 −0.99767 −0.63109 −0.81697 −0.73084 0.99767 11 0 0.2698 0.730840.99767 0.3984 −0.81697 −0.51958 0.97908 −0.3984 12 0 −0.38268 −0.83618−0.97547 −0.33488 0.76482 0.75371 −0.69493 −0.51958 13 0 0.068242−0.33488 0.73084 −0.99767 0.68255 0.33488 −0.99767 −2.3E−15 14 0−0.64424 −0.77571 0.99869 −0.99767 0.50492 0.96292 0.74238 0.94226 15 0−0.94784 0.99767 −0.69493 −0.81697 −0.50492 −0.88789 −0.47516 0.88789 160 −0.2862 −0.57668 0.414 0.2698 −0.8799 0.46007 −0.8799 0.2698 17 0−0.93641 0.70711 −0.95817 −0.96292 −0.78637 0.99942 −0.71908 −0.99767 180 −0.77571 0.2698 −0.96292 −0.73084 −0.91721 0.99767 −0.20346 −0.3984 190 −0.94226 0.91721 −0.20346 0.2698 0.51958 0.46007 0.20346 0.97908 20 0−0.13617 −1 −0.91721 0.88789 −0.88789 −0.99069 0.068242 −0.2698 21 0−0.83618 0.73084 −0.49011 0.13617 0.36685 0.88789 0.98546 −0.51958 22 0−0.97157 0.99069 0.90307 0.3984 0.23676 −0.06824 0.97157 −0.3984 23 0−0.99869 0.79681 0.86317 −0.91721 0.56265 0.034141 −0.2862 −0.97908 24 0−0.99287 −0.10226 −0.99869 0.33488 −0.71908 0.97157 0.44484 0.2698 25 00.46007 0.88789 −0.96292 −0.3984 −0.57668 −0.97908 0.068242 −0.97908 260 −0.94226 −0.97908 0.99767 0.73084 0.63109 0.2698 0.51958 −0.13617 27 00.95817 −0.54845 −0.86317 0.46007 −0.74238 −0.03414 −0.93641 0.94226 280 −0.61775 −0.03414 −0.94784 0.91721 −0.22014 0.23676 0.56265 0.3984 290 0.5341 −0.79681 0.8799 −0.85442 0.69493 −0.30251 −0.35092 0.94226Sequence n index 9 10 11 12 13 14 15 16 0 0.2698 −0.63109 0.73084−0.63109 0.2698 0.3984 −0.97908 0.51958 1 0.88789 0.3984 −0.13617 0.39840.88789 −0.99767 0.73084 0.81697 2 0.99767 −0.81697 −0.97908 −0.816970.99767 −0.63109 −0.94226 −0.13617 3 0.81697 −0.73084 0.2698 −0.730840.81697 0.13617 0.99767 −0.94226 4 0.63109 0.13617 0.88789 0.136170.63109 −0.51958 0.2698 −0.97908 5 0.13617 −0.94226 0.3984 −0.942260.13617 −0.97908 −0.63109 −0.2698 6 −0.13617 0.94226 −0.3984 0.94226−0.13617 0.97908 0.63109 0.2698 7 −0.63109 −0.13617 −0.88789 −0.13617−0.63109 0.51958 −0.2698 0.97908 8 −0.81697 0.73084 −0.2698 0.73084−0.81697 −0.13617 −0.99767 0.94226 9 −0.99767 0.81697 0.97908 0.81697−0.99767 0.63109 0.94226 0.13617 10 −0.88789 −0.3984 0.13617 −0.3984−0.88789 0.99767 −0.73084 −0.81697 11 −0.2698 0.63109 −0.73084 0.63109−0.2698 −0.3984 0.97908 −0.51958 12 0.99636 −0.54845 −0.11923 0.57668−0.78637 0.83618 −0.76482 0.51958 13 0.99069 0.46007 −0.51958 −0.97908−0.91721 −0.68255 −0.51958 −0.51958 14 0.61775 −0.99069 0.94784 −0.94226−0.2862 0.46007 0.38268 −0.51958 15 −0.64424 −0.63109 0.085266 −0.13617−0.97547 0.3984 −0.44484 0.99767 16 0.414 −0.57668 −0.2862 1.86E−140.99636 0.85442 0.98241 −0.51958 17 −0.8267 0.93028 −0.98822 −0.57668−0.84542 −0.23676 −0.01707 0.97908 18 0.85442 −0.81697 0.57668 0.51958−0.91721 0.97908 0.068242 0.3984 19 −0.88789 −0.85442 −0.96292 0.97908−0.3984 −0.33488 1 −0.81697 20 0.73084 0.068424 −0.57668 −0.63109−0.63109 −0.57668 0.068242 0.73084 21 0.99476 0.99767 −0.42948 0.942260.75371 0.63109 −0.23676 −0.81697 22 −0.42948 0.99069 −0.23676 −9.8E−150.6572 0.68255 0.95314 −0.73084 23 −0.11923 −0.30251 0.085266 −0.962920.18671 −0.10226 −0.414 −0.97908 24 0.76482 −0.97157 −0.99636 0.962920.94784 −0.97157 −0.99985 0.94226 25 0.20346 −0.99767 −0.20346 0.81697−0.91721 0.2698 −0.46007 0.88789 26 0.81697 −0.3984 0.88789 0.88789−0.3984 0.81697 −0.13617 0.51958 27 0.085266 0.79681 −0.56265 0.775710.38268 −0.87166 −0.25332 −0.3984 28 0.11923 0.99476 −0.93641 −0.77571−0.99287 0.93028 −0.56265 −0.51958 29 −0.74238 −0.42948 0.93641 0.46007−0.69493 −0.97157 −0.47516 0.13617 Sequence n index 17 18 19 20 21 22 230 0.81697 −0.3984 −0.99767 −0.73084 −0.2698 −4.4E−15 0 1 0.63109 0.99767−0.2698 0.13617 −0.88789 −1.8E−14 0 2 0.51958 0.63109 0.3984 0.97908−0.99767 1.95E−15 0 3 −0.97908 −0.13617 −0.51958 −0.2698 −0.81697−3.5E−14 0 4 0.73084 0.51958 −0.81697 −0.88789 −0.63109 −2.5E−14 0 5−0.88789 0.97908 0.73084 −0.3984 −0.13617 1.08E−13 0 6 0.88789 −0.97908−0.73084 0.3984 0.13617 3.91E−15 0 7 −0.73084 −0.51958 0.81697 0.887890.63109 2.35E−14 0 8 0.97908 0.13617 0.51958 0.2698 0.81697 1.47E−13 0 9−0.51958 −0.63109 −0.3984 −0.97908 0.99767 −6.1E−14 0 10 −0.63109−0.99767 0.2698 −0.13617 0.88789 1.86E−13 0 11 −0.81697 0.3984 0.997670.73084 0.2698 2.16E−13 0 12 −0.01707 −0.6572 0.99636 −0.33488 −0.845420.54845 0 13 −0.68255 −0.91721 −0.97908 −0.51958 0.46007 0.99069 0 14−0.71908 −0.96292 −0.80701 −0.2698 0.71908 0.85442 0 15 0.38268 0.26980.91028 0.13617 −0.414 −0.2698 0 16 0.50492 −0.99069 −0.80701 −0.97908−0.15306 0.46007 0 17 0.94784 −0.36685 0.35092 −0.96292 −0.99287−0.10226 0 18 0.68255 −0.2698 −0.96292 −0.63109 1 −0.73084 0 19 −0.99767−0.46007 0.57668 0.88789 0.99767 −0.57668 0 20 −0.2698 0.068242 −0.99069−0.88789 0.88789 −0.91721 0 21 0.30251 −0.3984 0.60424 −0.2698 0.034141−0.97908 0 22 0.54845 −0.20346 0.034141 −0.88789 −0.03414 −0.06824 0 23−0.71908 0.60424 0.96738 −0.33488 0.92388 0.93028 0 24 0.66998 −0.102260.61775 −1 −0.50492 −0.60424 0 25 1 −0.94226 −0.85442 0.81697 0.85442−0.94226 0 26 0.2698 0.63109 0.73084 0.99767 −0.97908 −0.94226 0 27−0.56265 −0.36685 −0.18671 −0.91721 0.50492 0.87166 0 28 0.97547−0.93028 −0.94784 −0.68255 0.78637 0.70711 0 29 0.5341 0.70711 0.719080.57668 0.22014 −0.36685 0

Below table 9 shows the comparison between the sequences generated bythe proposed generating equation and the ZC sequences generated byapplying the conventional cyclic extension when N=24.

TABLE 9 Conventional ZC Proposed sequence Sequence Num. of Sequences 2230 Num. of Sequences < CM 1.2 dB 12 30 Max. CM [dB] 2.01 1.14 Max.Cross. Cor. 0.36 0.39 Mean Cross. Cor. 0.19 0.18 Median Cross. Cor. 0.200.18 Std. Cross. Cor. 0.07 0.09

It is noted that when the sequences are generated according to theproposed method, the number of available sequences is increased andbetter cross-correlation characteristics are obtained. When frequencyhopping in an actual environment is counted, a BLER performance becomesbetter as a mean correlation value is lower. Thus, the BLER performanceof the proposed sequences is superior.

V. Order Restriction of Phase Equation

The relation between the order ‘k’ of a phase equation with respect to aphase component of a sequence, the number of available sequences, andthe correlation characteristics is as follows.

As the order ‘k’ is increased, the number of available sequences isincreased but the correlation characteristics are degraded. As the order‘k’ becomes small, the number of available sequences is reduced but thecorrelation characteristics are improved. If k=2, ZC sequences can begenerated, so if k>2, a restriction for generation of sequences isrequired.

A method for restricting the order of a phase equation according to thelength of desired sequences according to the desired number of availablesequences in consideration of the number of available sequences andcorrelation characteristics, when a third or hither polynomialexpression is applied to phase components of sequences will now bedescribed. When the desired number of minimum available sequences isNseq, if the number of sequences (Nposs) that can be generated by usingthe second order phase equation with a desired length N of sequences islarger than or the same as Nseq (i.e. Nposs>=Nseq), the second orderphase equation is used. If Nposs<Nseq, a third or higher order phaseequation is used.

It can be expressed by stages as follows:

Step 1: The desired number Nseq of minimum available sequences isdetermined

Step 2: The number Nposs of available sequences that can be generated bythe second order phase equation (k=2) is determined from the length N ofthe desired sequences.

Step 3: If Nposs is larger than or the same as Nseq, sequences aregenerated by using the second order phase equation, and if Nposs issmaller than Nseq, sequences are generated by using the third orderphase equation.

First Embodiment

The following sequence generating equation having the third phaseequation with k=3 is considered:

$\begin{matrix}{{{r(n)} = {{\mathbb{e}}^{{j\alpha}\; n}{x_{u}\left( {\left( {n + \theta} \right){{mod}N}_{ZC}} \right)}}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{b_{0}u_{0}m^{3}} + {b_{1}u_{1}m^{2}} + {u_{2}m}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 29} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N_(ZC)−1, N is the length of the sequence r(n),u₀, u₁, . . . , u_(k-1) are arbitrary integers, ‘a’ is an arbitrary realnumber, and N_(ZC) is the largest prime number among natural numberssmaller than N. e^(jan) is an expression, in the frequency domain, ofperforming cyclic shift by ‘α’ in the time domain. θ is a shift offsetvalue and indicates performing of cyclic extension after shifting by θ.

It is assumed that the length N of the desired sequences is possible inthe following case:

N=[12 24 36 48 60 72 96 108 120 144 180 192 216 240 288 300]

In step 1, the number Nseq of minimum available sequences is set to 30.In step 2, if the second phase equation is a=1, u₀=0, u₁=u₂=u, b₀=0, andb₁=b₂=1 in Equation 29, the available number Nposs of available ZCsequences of each N is as follows:

Nposs=[10 22 30 46 58 70 88 106 112 138 178 190 210 238 282 292]

In step 3, the length of sequences that can use the second order phaseequation is N=[36 48 60 72 96 108 120 144 180 192 216 240 288 300], andthe length of sequences that can use the third order phase equation isN=[12 24].

Second Embodiment

The following sequence generating equation having the third phaseequation with k=3 is considered.

$\begin{matrix}{{{r(n)} = {{\mathbb{e}}^{{j\alpha}\; n}{x_{u}\left( {\left( {n + \theta} \right){mod}\; N_{ZC}} \right)}}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{2u_{0}m^{3}} + {u_{1}m^{2}} + {u_{2}m}})}}}{N_{ZC}}}}} & \left\lbrack {{Equation}\mspace{14mu} 30} \right\rbrack\end{matrix}$

It is assumed that the length N of desired sequences is possible in thefollowing case:

N=[12 24 36 48 60 72 96 108 120 144 180 192 216 240 288 300]

In step 1, the number Nseq of minimum available sequences is 30. In step2, if the second order phase equation is a=1, u₀=0, and u₁=u₂=u inEquation 30, the available number Nposs of available ZC sequences ofeach N is as follows:

Nposs=[10 22 30 46 58 70 88 106 112 138 178 190 210 238 282 292]

In step 3, the length of sequences that can use the second order phaseequation is N=[36 48 60 72 96 108 120 144 180 192 216 240 288 300], andthe length of sequences that can use the third order phase equation isN=[12 24].

The sequence generating equation for which the order of the phaseequation is restricted can be expressed by two types. In a firstexpression method, it is assumed that a sequence with a length N ismapped in the frequency domain. This means that each element of thesequence is mapped to the N number of subcarriers. First, it is assumedthat the sequence r(n) is given as follows.r(n)=e ^(jαn) x _(u)((n+θ)modN _(ZC)), 0≦n≦N−1  [Equation 31]

According to the first type of sequence generating equation, when thelength N of sequences is larger than or the same as 36, a base sequencex_(u)(m) is given as follows:

$\begin{matrix}{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\; u\;{m{({m + 1})}}}{N_{ZC}}}} & \left\lbrack {{Equation}\mspace{14mu} 32} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N_(ZC)−1.

If the length N of the sequences is smaller than 36, the base sequencex_(u)(m) is given as follows.

$\begin{matrix}{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{{\pi 0}{.125}{({{2u_{0}m^{3}} + {u_{1}m^{2}} + {u_{2}m}})}}{N_{ZC}}}} & \left\lbrack {{Equation}\mspace{14mu} 33} \right\rbrack\end{matrix}$

According to a second type of sequence generating equation, the basesequence x_(u)(m) is given as follows:

$\begin{matrix}{{x_{u}(m)} = {\mathbb{e}}^{{- j}\frac{\pi\;{a{({{2u_{0}m^{3}} + {u_{1}m^{2}} + {u_{2}m}})}}}{N_{ZC}}}} & \left\lbrack {{Equation}\mspace{14mu} 34} \right\rbrack\end{matrix}$

where when the length N of sequences is larger than or the same as 36,a=1 and u₁=u₂=u, and if the length N of sequences is smaller than 36, ifa=0.125 and N=12, u₁ and u₂ are defined by the below Table 10.

TABLE 10 Sequence Index u₁ u₂ u₃ CM [dB] 1 0 8 8 0.17 2 0 32 32 0.85 3 040 40 0.43 4 0 48 48 0.43 5 0 56 56 0.85 6 0 80 80 0.17 7 0 19 10 1.08 80 26 0 1.12 9 0 61 0 0.87 10 0 68 3 1.18 11 1 78 22 1.11 12 2 25 60 0.9913 3 62 2 1.15 14 3 73 4 1.15 15 3 80 37 1.10 16 4 82 8 1.18 17 11 38 861.18 18 12 65 75 1.12 19 14 73 52 1.20 20 16 83 61 1.05 21 18 34 11 1.1122 18 50 41 1.16 23 22 17 44 0.88 24 25 61 36 1.14 25 25 88 11 1.17 2627 39 5 1.12 27 32 23 85 1.12 28 34 17 52 1.10 29 38 36 31 1.04 30 40 68 1.18

If N=24, u₁ and u₂ are defined by the below Table 11.

TABLE 11 Sequence Index u₁ u₂ u₃ 1 0 8 8 2 0 32 32 3 0 48 48 4 0 64 64 50 72 72 6 0 88 88 7 0 96 96 8 0 112 112 9 0 120 120 10 0 136 136 11 0152 152 12 0 176 176 13 0 6 17 14 0 6 182 15 0 25 16 16 0 29 82 17 0 35132 18 0 44 27 19 0 48 4 20 0 54 18 21 0 54 122 22 0 58 0 23 0 64 14 240 68 21 25 0 88 11 26 0 96 116 27 0 112 0 28 0 126 133 29 0 130 15 30 0178 39

VI. Generation of Sequences for a Reference Signal

The following sequence generating equation is considered:

$\begin{matrix}{{{r(n)} = {{\mathbb{e}}^{{j\alpha}\; n}{x_{u}\left( {\left( {n + \theta} \right){mod}\; N_{ZC}} \right)}}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- {j\pi}}{\{{{{quan}(\frac{a{({{u_{0}m^{3}} + {u_{1}m^{2}} + {u_{2}m^{1}}})}}{N_{ZC}})} + u_{3}}\}}}}} & \left\lbrack {{Equation}\mspace{14mu} 35} \right\rbrack\end{matrix}$

where m=0, 1, . . . , N_(ZC)−1, a=0.0625, u3=¼, N is the length of thesequence r(n), u₀, u₁, and u₂ are arbitrary integers, θ is a shiftoffset value, and N_(ZC) is the largest prime number among naturalnumbers smaller than N. The Quantization function quan(.) isapproximated to the closest {0, ½, 1, 3/2, 2, . . . }. Namely, thequantization function quan(x) is approximated to an integer orinteger+0.5 closest to ‘x’. It can be expressed by quan(x)=round(2x)/2,and round(x) is an integer immediately smaller than x+0.5.

A memory capacity can be saved through quantization. The range of u₀,u₁, and u₂ may be extended to increase the degree of freedom to therebygenerate a larger number of sequences with good performance. In thisrespect, however, the increase in the range of u₀, u₁, and u₂ causes anincrease in the number of bits to represents u₀, u₁, and u₂. Thus, it isrestricted with QPSK modulation so that only two bits are required pervalue regardless of the range of u₀, u₁, and u₂. In addition, becausethe basic generating equation is based on the CAZAC sequence, sequenceswith good correlation characteristics can be generated. For example, ifthe range of 0≦u₀≦1024, 0≦u₁≦1024, and 0≦u₂≦1024 is provided to generatesequences of a length of 12, memory of 30 bits (=10 bits+10 bits+10bits) is used per sequence, so 900 bits of memory capacity is requiredfor 30 sequences. However, when quantization is performed, memory of 720bits (=2 bits×12×30) is sufficient for sequence regardless of the rangeof u₀, u₁, and u₂.

The above generating equation may be equivalent to a value obtained byapproximating elements of sequences to a QPSK constellation phase. Thisis because every value can be approximated with the Nq number of valuesquantized between 0 and 2π that may expressed by phases throughquantization function. Namely, values in a complex unit circuit thee^(−jθ) may have can be quantized to the Nq number of values to therebyapproximate every value.

In this case, according to the approximating methods, the values can beapproximated to the closest values, to the same or the closest smallvalues, or to the same or the closest large values.

Elements of sequences can be approximated to values of {π/4, 3π/4, −π/4,−3π/4} corresponding to the phases of QPSK. This means that thequantized values are approximated to the coordinates of QPSK{0.7071+j0.7071, 0.7071−j0.7071, −0.7071+j0.7071, −0.7071−j0.7071}.

Hereinafter, generation of extended sequence will be described, but atruncated sequence as in the following equation may be also usedaccording to the length N of the desired sequences and the length N_(ZC)of the ZC sequences.

$\begin{matrix}{{{r(n)} = {{\mathbb{e}}^{{j\alpha}\; n}{x_{u}(n)}}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- {j\pi}}{\{{{{quan}(\frac{a{({{u_{0}m^{3}} + {u_{1}m^{2}} + {u_{2}m^{1}}})}}{N_{Zc}})} + u_{3}}\}}}}} & \left\lbrack {{Equation}\mspace{14mu} 36} \right\rbrack\end{matrix}$

Alternatively, if the length N of the desired sequences and the lengthN_(ZC) of the ZC sequences are the same, sequences as in the followingequation may be also used.

$\begin{matrix}{{{r(n)} = {{\mathbb{e}}^{{j\alpha}\; n}{x_{u}(n)}}},{0 \leq n \leq {N - 1}},{{x_{u}(m)} = {\mathbb{e}}^{{- {j\pi}}{\{{{{quan}(\frac{a{({{u_{0}m^{3}} + {u_{1}m^{2}} + {u_{2}m^{1}}})}}{N})} + u_{3}}\}}}}} & \left\lbrack {{Equation}\mspace{14mu} 37} \right\rbrack\end{matrix}$

Substantial examples to generate a sequence generation for a referencesignal will now be described.

In the uplink subframe, a PUCCH or a PUSCH is scheduled by a unit ofresource blocks, and a resource block includes twelve subcarriers. Thus,a sequence with a length N=12 is required for a single resource block, asequence of with a length N=24 is required for two resource blocks. Thesequence with the length N=12 may be generated by cyclic-extending asequence with N_(ZC)=11, and the sequence with the length N=24 may begenerated by cyclic-extending a sequence with N_(ZC)=23.

(1) Reference Signal Sequence for N=12

The below table shows u₀, u₁, and u₂, when N=12. It shows 30 sequencecombinations, which do not have such a high cross-correlation withextended ZC sequences corresponding to three resource blocks, assearched from sequences that do not exceed a CM 1.22 dB, bypreferentially considering a CP (Cyclic Prefix) as the CM.

TABLE 12 index (u) u₀ u₁ u₂ 0 29995 30337 2400 1 32762 2119 36039 235746 37587 26527 3 18603 33973 25011 4 18710 2129 19429 5 5033 2814514997 6 6940 23410 7920 7 19235 26638 38189 8 2037 29 16723 9 8965 2979525415 10 35666 2400 4229 11 7660 31762 17023 12 23501 14111 6290 1332271 14654 3822 14 16265 29599 640 15 26931 38734 3401 16 11963 2970622674 17 9560 24757 22880 18 22707 14318 7654 19 16440 14635 3587 2022209 13004 10470 21 23277 2967 19770 22 25054 36961 9673 23 39007 3698421639 24 5353 38653 26803 25 36686 19758 36923 26 3768 37064 30757 2715927 15525 13082 28 33614 17418 37090 29 33995 7240 12053

The reference signal sequence r(n) with the length 12 generated from theabove table can be expressed by the following equation:r(n)=e ^(jαn) x _(u(n)) , x _(u)(n)=e ^(p(n)π/4), 0≦n<N  [Equation 38]

Where ‘α’ is a cyclic shift value, and values of the phase parametersp(n) of the base sequences x_(u)(n) are given as shown in the followingtable:

TABLE 13 Sequence index (u) p(0), . . . , p(11) 0 −1 −3 −1 1 1 −3 3 1 31 1 −1 1 −1 3 3 −3 3 1 −1 −1 −3 −1 1 −1 2 −1 1 −3 −1 −3 −3 −3 1 −1 −3 1−1 3 −1 3 −1 1 1 −3 −3 −1 −3 −3 3 −1 4 −1 3 1 −3 3 −3 −1 −3 −3 3 −3 −1 5−1 1 3 3 1 −3 3 3 1 3 −1 −1 6 −1 1 1 3 −1 1 1 1 −1 −3 3 −1 7 −1 1 1 −1−1 −1 3 1 3 −3 3 −1 8 −1 3 1 3 1 −1 −1 3 −3 −1 −3 −1 9 −1 −3 1 −1 −3 1 11 −1 1 −3 −1 10 −1 −3 −1 −3 −1 −1 3 −3 −3 3 1 −1 11 −1 −3 1 1 −1 1 −1 −13 −3 −3 −1 12 −1 1 −3 −1 1 −1 3 3 1 −1 1 −1 13 −1 −3 1 3 −1 −1 3 1 1 1 1−1 14 −1 −1 1 1 −1 −3 −1 −3 −3 1 3 −1 15 −1 −3 3 −1 1 1 −3 −1 −3 −1 −1−1 16 −1 1 −3 3 −1 −1 3 −3 −3 −3 −3 −1 17 −1 3 3 −3 3 −3 −3 1 1 −1 −3 −118 −1 −1 −3 −1 −3 −3 1 1 3 −3 3 −1 19 −1 3 −3 1 −1 3 −1 −3 −3 −1 −1 −120 −1 1 −1 −3 −1 −1 1 3 −3 3 3 −1 21 −1 1 −1 1 1 3 −1 −1 −3 3 3 −1 22 −11 3 −3 −3 3 −1 −1 −3 −1 −3 −1 23 −1 −3 1 −1 −3 −1 3 −3 3 −3 −1 −1 24 −1−3 −3 −3 1 −1 1 1 −3 −1 1 −1 25 −1 −3 −1 1 1 3 −1 1 −1 −3 3 −1 26 −1 3 31 −3 −3 −1 1 1 −1 1 −1 27 −1 3 1 −1 −3 −3 −3 −1 3 −3 −3 −1 28 −1 −3 1 11 1 3 1 −1 1 −3 −1 29 −1 3 −3 3 −1 3 3 −3 3 3 −1 −1

(2) Reference Signal Sequence for N=24

The below table shows u₀, u₁, and u₂, when N=12. It shows 30 sequencecombinations, which do not have such a high cross-correlation withextended ZC sequences corresponding to three resource blocks, assearched from sequences that do not exceed a CM 1.22 dB, bypreferentially considering a CP (Cyclic Prefix) as the CM.

TABLE 14 Index (u) u₀ u₁ u₂ 0 35297 9057 9020 1 24379 861 26828 2 158964800 31943 3 6986 9180 7583 4 22605 15812 10886 5 852 3220 18552 6 1604810573 27569 7 15076 9412 26787 8 15074 3760 38376 9 38981 11775 37785 1029686 14549 13300 11 21429 7431 34668 12 28189 33097 5721 13 6551 3469436165 14 25833 17562 20508 15 38286 20581 17410 16 17305 10299 10752 1727571 8218 1477 18 16602 31085 15253 19 14199 11732 25429 20 1665 941524015 21 33837 26684 9587 22 20569 33119 21324 23 27246 33775 21065 2418611 30085 28779 25 29485 39582 28791 26 21508 25272 21422 27 595625772 2113 28 17823 13894 23873 29 5862 3810 35855

The reference signal sequence r(n) with the length 24 generated from theabove table can be expressed by the following equation:r(n)=e ^(jαn) x _(u(n)) , x _(u)(n)=e ^(p(n)π/4), 0≦n<N  [Equation 39]

Where ‘α’ is a cyclic shift value, and values of the phase parametersp(n) of the base sequences x_(u)(n) are given as shown in the followingtable:

TABLE 15 Seq. index(u) p(0), . . . , p(23) 0 −1 3 3 3 −3 1 3 −3 −1 −3 −11 −3 3 −1 3 1 1 −1 −3 3 −1 −1 −1 1 −1 1 3 −3 3 −1 3 −1 1 −1 −3 −1 3 −3−1 −3 −3 −3 −3 −1 −1 3 1 −1 2 −1 3 1 −1 −3 1 −3 3 1 1 −1 −3 −1 1 −3 −1 11 1 3 −3 −1 −1 −1 3 −1 −3 3 3 −3 1 −1 −1 3 3 −3 −3 1 3 3 −1 3 −3 −1 −1−1 −1 1 −1 4 −1 −1 −3 −1 3 −3 −3 −1 3 −3 3 1 1 −3 −3 −3 −3 1 −1 −3 1 −1−1 −1 5 −1 1 −3 −3 −1 −1 1 −1 −3 −3 −3 3 1 −3 1 3 1 −3 3 −3 −1 −3 −3 −16 −1 1 1 3 −3 −1 1 −1 −1 −1 3 1 −1 1 −3 1 3 −3 3 1 −3 1 1 −1 7 −1 1 1 −3−3 −3 −3 −3 −3 −1 3 3 −1 −1 −3 1 −3 1 −3 1 1 −3 3 −1 8 −1 −1 3 3 −1 3 1−3 −3 1 −3 −1 3 −1 −1 −1 −3 1 1 −1 −3 −3 −3 −1 9 −1 −3 3 −1 −1 −1 −1 1 1−3 3 1 3 3 1 −1 1 −3 1 −3 1 1 −3 −1 10 −1 −3 −1 −3 −1 −3 −3 1 1 3 1 3 −1−1 3 1 1 −3 −3 −1 3 3 −1 −1 11 −1 −3 3 −3 −3 −3 −1 −1 −3 −1 −3 3 1 3 −3−1 3 −1 1 −1 3 −3 1 −1 12 −1 −1 1 −3 1 3 −3 1 −1 −3 −1 3 1 3 1 −1 −3 −3−1 −1 −3 −3 −3 −1 13 −1 −3 −1 −1 3 1 3 1 −3 −3 −1 −3 −3 −3 −1 3 3 −1 −1−3 1 3 −1 −1 14 −1 1 −1 −1 3 −3 3 −3 1 1 −1 1 1 1 −1 3 3 −1 −1 1 −3 3 −1−1 15 −1 1 −3 −3 −3 1 1 −3 1 1 −1 −1 3 −1 −3 1 −1 −1 1 1 3 1 −3 −1 16 −1−1 −3 3 −1 −1 −1 3 1 −3 1 1 3 −3 1 −3 −1 −1 −1 3 −3 3 −3 −1 17 −1 1 −3−1 −3 1 3 −3 3 −3 −3 −3 1 −1 3 −1 −3 −1 −1 −3 −3 1 1 −1 18 −1 3 1 −3 −3−3 −3 1 −1 1 1 1 −3 −1 1 1 3 −1 −1 3 −1 1 −3 −1 19 −1 1 −3 −1 −1 1 −3 −1−3 −1 1 1 1 1 3 1 −1 −3 −3 3 −1 3 1 −1 20 −1 1 3 −1 −1 1 −1 −3 −1 −1 1 11 −3 3 1 −1 −1 −3 3 −3 −1 3 −1 21 −1 −3 1 1 3 −3 1 1 −3 −1 −1 1 3 1 3 1−1 3 1 1 −3 −1 −3 −1 22 −1 −1 3 3 3 −3 −3 3 3 −1 3 −1 −1 −1 −1 3 −3 1 −13 −1 −1 3 −1 23 −1 3 −1 3 −1 1 1 3 1 3 −3 1 3 −3 −3 1 1 −3 3 3 3 1 −1 −124 −1 −3 −1 −1 1 −3 −1 −1 1 −1 −3 1 1 −3 1 −3 −3 3 1 1 −1 3 −1 −1 25 −1−1 1 −1 1 1 −1 −1 −3 1 −3 −1 3 1 −3 3 −1 1 3 −3 3 1 −1 −1 26 −1 1 −3 −3−1 1 −1 1 3 1 3 3 1 1 −1 3 1 −1 3 −1 3 3 −1 −1 27 −1 −1 3 −3 1 −3 1 3 −33 1 3 3 −1 −1 3 3 3 1 −3 −1 −1 1 −1 28 −1 3 −1 −1 3 −3 −1 −3 −3 3 −3 3−1 1 −1 −3 3 3 −3 −1 −1 −1 1 −1 29 −1 1 1 1 3 3 −1 −1 −1 −1 3 −3 −1 3 1−3 −1 1 −1 −1 −3 1 −1 −1

VII. Selection of Sequence for Reference Signal

In the above description, the sequences are generated from theclosed-form generation equation with respect to N=12 and N=24. However,in an actual wireless communication system, sequences generated from asingle generating equation may not be applicable but mixed with othersequences. Thus, correlation characteristics or CM characteristicsbetween the thusly generated sequences and other sequences need to beconsidered.

Here, a method, in which 30 sequences generated from Equation 38 andTable 13 when N=12 are compared with 26 comparative sequences and foursequences with good correlation characteristics are selected asreference signal sequences, will now be described. Further, a method, inwhich 30 sequences generated from Equation 39 and Table 15 when N=24 arecompared with 25 comparative sequences and five sequences with goodcorrelation characteristics are selected as reference signal sequences,will now be described.

(1) In Case of N=12

If N=12, a sequence generating equation is a cyclic shift of the basesequence x_(u)(n) like Equation 38, and values of the phase parametersp(n) of the base sequences x_(u)(n) are given as those shown in Table13. Here, the method, in which 30 sequences generated when N=12 arecompared with 26 comparative sequences and four sequences with goodcorrelation characteristics are selected, will now be described. Thenumber of cases of choosing four base sequences from among 30 basesequences is 27405 (₃₀C₄=30*29*28*27/4/3/2/1=27405). Thus, in order toreduce the number of cases, first, CM of the base sequences isconsidered.

The below table shows base sequences arranged in the order of CM size.In the table, the largest value among the CM values of all the possiblecyclic shifts of the base sequences is determined as a representativeCM.

TABLE 16 Sequence Index CM 23 0.6486 26 0.6634 29 0.8258 21 0.8961 150.9052 12 0.9328 14 0.977 28 0.9773 19 0.987 25 0.9991 1 1.0015 5 1.001922 1.0273 11 1.035 20 1.0376 18 1.0406 10 1.0455 3 1.05 0 1.0608 171.066 8 1.073 24 1.0927 9 1.1054 2 1.1054 4 1.1248 27 1.1478 6 1.1478 161.1502 7 1.1616 13 1.1696

When N=12, namely, because the length of base sequences corresponding toa single resource block is short, many sequences have similarcross-correlation characteristics, so sequences with a CM of more than acertain value are excluded. Here, sequences [23 29 21 15 12 14 28 19 251 5 22 11 20 18 10 3 0 17 8] having a CM lower than 1.09 are considered.

It is assumed that phase parameters p^(c)(n) of comparative sequencesthat can be used together with the base sequences are those as shown inthe below table. In this case, the comparative sequences are differentin their phase parameters but the same in their forms as the basesequences.

TABLE 17 Comparative Sequence Index p^(c)(0), . . . , p^(c)(11) 0 −1 1 3−3 3 3 1 1 3 1 −3 3 1 1 1 3 3 3 −1 1 −3 −3 1 −3 3 2 1 1 −3 −3 −3 −1 −3−3 1 −3 1 −1 3 −1 1 1 1 1 −1 −3 −3 1 −3 3 −1 4 −1 3 1 −1 1 −1 −3 −1 1 −11 3 5 1 −3 3 −1 −1 1 1 −1 −1 3 −3 1 6 −1 3 −3 −3 −3 3 1 −1 3 3 −3 1 7 −3−1 −1 −1 1 −3 3 −1 1 −3 3 1 8 1 −3 3 1 −1 −1 −1 1 1 3 −1 1 9 1 −3 −1 3 3−1 −3 1 1 1 1 1 10 3 1 −1 −1 3 3 −3 1 3 1 3 3 11 1 −3 1 1 −3 1 1 1 −3 −3−3 1 12 3 3 −3 3 −3 1 1 3 −1 −3 3 3 13 −3 1 −1 −3 −1 3 1 3 3 3 −1 1 14 3−1 1 −3 −1 −1 1 1 3 1 −1 −3 15 1 3 1 −1 1 3 3 3 −1 −1 3 −1 16 −3 1 1 3−3 3 −3 −3 3 1 3 −1 17 −3 3 1 1 −3 1 −3 −3 −1 −1 1 −3 18 −1 3 −1 1 −3 −3−3 −3 −3 1 −1 −3 19 1 1 −3 −3 −3 −3 −1 3 −3 1 −3 3 20 1 1 −1 −3 −1 −3 1−1 1 3 −1 1 21 1 1 3 1 3 3 −1 1 −1 −3 −3 1 22 1 −3 3 3 1 3 3 1 −3 −1 −13 23 1 3 −3 −3 3 −3 1 −1 −1 3 −1 −3 24 −3 −1 −3 −1 −3 3 1 −1 1 3 −3 −325 3 −3 −3 −1 −1 −3 −1 3 −3 3 1 −1

Of the 30 base sequences, the best 25 combinations among the maximumcross correlation combinations with the comparative sequences, are thoseas shown in the below table.

TABLE 18 Combination of Mean Max No. Sequence Indexes CorrelationCorrelation 1 0 3 8 17 0.2568 0.644 2 3 8 17 25 0.2567 0.6546 3 0 8 1725 0.2567 0.6546 4 0 3 17 25 0.2576 0.6546 5 0 3 8 25 0.2561 0.6546 6 817 25 28 0.2568 0.6546 7 3 17 25 28 0.2576 0.6546 8 0 17 25 28 0.25770.6546 9 3 8 25 28 0.2561 0.6546 10 0 8 25 28 0.2562 0.6546 11 0 3 25 280.2571 0.6546 12 3 8 17 28 0.2568 0.6546 13 0 8 17 28 0.2569 0.6546 14 03 17 28 0.2577 0.6546 15 0 3 8 28 0.2562 0.6546 16 17 25 28 29 0.25760.6755 17 8 25 28 29 0.2561 0.6755 18 3 25 28 29 0.257 0.6755 19 0 25 2829 0.257 0.6755 20 8 17 28 29 0.2568 0.6755 21 3 17 28 29 0.2576 0.675522 0 17 28 29 0.2577 0.6755 23 3 8 28 29 0.2560 0.6755 24 0 8 28 290.2562 0.6755 25 0 3 28 29 0.2571 0.6755

From the above table, if four sequences that have good mean crosscharacteristics and maximum cross characteristics when compared with thecomparative sequences and satisfy desired CM characteristics are to beselected from among the 30 sequences having the same base sequencegenerating equation as the Equation 36 and having the values of thephase parameters p(n) as provided in Table 13, the four sequences havingthe sequence indexes [3 8 28 29] would be base sequences.

Finally, the reference signal sequence r(n) with the length N=12 is asfollows:r(n)=e ^(jαn) x _(u)(n), 0≦n<N x _(u)(n)=e ^(jp(n)π/4)  [Equation 40]

where ‘α’ is a cyclic shift value, and the values of the phaseparameters p(n) of the base sequences x_(u)(n) are given as those shownin the below table.

TABLE 19 p(0), . . . , p(11) −1 3 −1 1 1 −3 −3 −1 −3 −3 3 −1 −1 3 1 3 1−1 −1 3 −3 −1 −3 −1 −1 −3 1 1 1 1 3 1 −1 1 −3 −1 −1 3 −3 3 −1 3 3 −3 3 3−1 −1

(2) In Case of N=24

When N=12, a sequence generating equation is a cyclic shift of the basesequence x_(u)(n) like Equation 37, and values of the phase parametersp(n) of the base sequences x_(u)(n) are given as those shown in Table15. Here, the method, in which the 30 sequences generated when N=24 arecompared with 25 comparative sequences and five sequences with goodcorrelation characteristics are selected, will now be described. Thenumber of cases of choosing five base sequences from among 30 basesequences is 142506 (₃₀C₄=30*29*28*27*26/5/4/3/2/1=142506).

It is assumed that phase parameters p^(c)(n) of the comparativesequences that can be used together with the base sequences are those asshown in the below table. In this case, the comparative sequences aredifferent only in their phase parameters but the same in their forms asthe base sequences.

TABLE 20 Comp. Sequence Index p^(c)(0), . . . , p^(c)(23) 0 −1 3 1 −3 3−1 1 3 −3 3 1 3 −3 3 1 1 −1 1 3 −3 3 −3 −1 −3 1 −3 3 −3 −3 −3 1 −3 −3 3−1 1 1 1 3 1 −1 3 −3 −3 1 3 1 1 −3 2 3 −1 3 3 1 1 −3 3 3 3 3 1 −1 3 −1 11 −1 −3 −1 −1 1 3 3 3 −1 −1 −1 −3 −3 −1 1 1 3 3 −1 3 −1 1 −1 −3 1 −1 −3−3 1 −3 −1 −1 4 −3 1 1 3 −1 1 3 1 −3 1 −3 1 1 −1 −1 3 −1 −3 3 −3 −3 −3 11 5 1 1 −1 −1 3 −3 −3 3 −3 1 −1 −1 1 −1 1 1 −1 −3 −1 1 −1 3 −1 −3 6 −3 33 −1 −1 −3 −1 3 1 3 1 3 1 1 −1 3 1 −1 1 3 −3 −1 −1 1 7 −3 1 3 −3 1 −1 −33 −3 3 −1 −1 −1 −1 1 −3 −3 −3 1 −3 −3 −3 1 −3 8 1 1 −3 3 3 −1 −3 −1 3 −33 3 3 −1 1 1 −3 1 −1 1 1 −3 1 1 9 −1 1 −3 −3 3 −1 3 −1 −1 −3 −3 −3 −1 −3−3 1 −1 1 3 3 −1 1 −1 3 10 1 3 3 −3 −3 1 3 1 −1 −3 −3 −3 3 3 −3 3 3 −1−3 3 −1 1 −3 1 11 1 3 3 1 1 1 −1 −1 1 −3 3 −1 1 1 −3 3 3 −1 −3 3 −3 −1−3 −1 12 3 −1 −1 −1 −1 −3 −1 3 3 1 −1 1 3 3 3 −1 1 1 −3 1 3 −1 −3 3 13−3 −3 3 1 3 1 −3 3 1 3 1 1 3 3 −1 −1 −3 1 −3 −1 3 1 1 3 14 1 3 −1 3 3 −1−3 1 −1 −3 3 3 3 −1 1 1 3 −1 −3 −1 3 −1 −1 −1 15 1 1 1 1 1 −1 3 −1 −3 11 3 −3 1 −3 −1 1 1 −3 −3 3 1 1 −3 16 1 3 3 1 −1 −3 3 −1 3 3 3 −3 1 −1 1−1 −3 −1 1 3 −1 3 −3 −3 17 −3 −3 1 1 −1 1 −1 1 −1 3 1 −3 −1 1 −1 1 −1 −13 3 −3 −1 1 −3 18 −3 −1 −3 3 1 −1 −3 −1 −3 −3 3 −3 3 −3 −1 1 3 1 −3 1 33 −1 −3 19 −1 −1 −1 −1 3 3 3 1 3 3 −3 1 3 −1 3 −1 3 3 −3 3 1 −1 3 3 20 1−1 3 3 −1 −3 3 −3 −1 −1 3 −1 3 −1 −1 1 1 1 1 −1 −1 −3 −1 3 21 1 −1 1 −13 −1 3 1 1 −1 −1 −3 1 1 −3 1 3 −3 1 1 −3 −3 −1 −1 22 −3 −1 1 3 1 1 −3 −1−1 −3 3 −3 3 1 −3 3 −3 1 −1 1 −3 1 1 1 23 −1 −3 3 3 1 1 3 −1 −3 −1 −1 −13 1 −3 −3 −1 3 −3 −1 −3 −1 −3 −1 24 1 1 −1 −1 −3 −1 3 −1 3 −1 1 3 1 −1 31 3 −3 −3 1 −1 −1 1 3

20 combinations with the best cross-correlation characteristics amongall the possible combinations are those as shown in the below table.

TABLE 21 Combination of Mean Max No. Sequence Indexes CorrelationCorrelation 1 9 11 16 21 27 0.1811 0.4791 2 11 12 16 21 25 0.181 0.48443 9 12 16 21 25 0.181 0.4844 4 9 11 12 21 25 0.1812 0.4844 5 9 11 12 1625 0.1812 0.4844 6 9 11 12 16 25 0.1811 0.4844 7 12 16 21 24 25 0.18060.4917 8 11 16 21 24 25 0.1808 0.4917 9 9 16 21 24 25 0.1807 0.4917 1011 12 21 24 25 0.1808 0.4917 11 9 12 21 24 25 0.1807 0.4917 12 9 11 2124 25 0.189 0.4917 13 11 12 16 24 25 0.1809 0.4917 14 9 12 16 24 250.1808 0.4917 15 9 11 16 24 25 0.181 0.4917 16 9 11 12 24 25 0.1810.4917 17 11 12 16 21 24 0.1807 0.4917 18 9 12 16 21 24 0.1806 0.4917 199 11 16 21 24 0.1808 0.4917 20 9 11 12 21 24 0.1808 0.4917

Among the combinations, combinations {7, 8, 9, 10, 11, 12, 13, 14, 17,18, 19, 20} have a mean correlation value greater than 0.181.

The below table shows base sequences arranged in the order of CM size.In the table, the largest value among the CM values of all the possiblecyclic shifts of the base sequences is determined as a representativeCM.

TABLE 22 Sequence Index CM 6 0.6423 12 0.7252 23 0.7632 20 0.8265 80.883 9 0.8837 19 0.9374 10 0.966 25 0.9787 11 0.9851 13 0.9966 291.0025 14 1.0112 28 1.0113 27 1.0143 17 1.0176 7 1.0191 22 1.0316 241.0387 5 1.0407 18 1.059 15 1.0722 3 1.0754 0 1.0761 21 1.094 1 1.095216 1.1131 26 1.1193 4 1.1223 2 1.1251

Sequence indexes included in the selected combinations are 9, 11, 12,16, 21, 24, 25, of which the sequence index 16 is excluded because ithas low CM characteristics of the base sequence. Thus, the selectablecombinations are reduced to the following four sequence indexes.

TABLE 23 Combination of Mean Max Sequence Indexes CorrelationCorrelation 11 12 21 24 25 0.1808 0.4917 9 12 21 24 25 0.1807 0.4917 911 12 21 24 0.1806 0.4917 9 11 21 24 25 0.1809 0.4917

If five sequences, which have good cross-correlation characteristics andCM characteristics with the comparative sequences and have a minimumcorrelation values, are to be selected from the above combinations, thesequences [9 11 12 21 24] will be base sequences.

Finally, the reference signal sequence r(n) with the length N=24 is asfollows:r(n)=e ^(jαn) x _(u)(n), 0≦n<N x _(u)(n)=e ^(jp(n)π/4)  [Equation 41]

wherein ‘α’ is a cyclic shift value, and the values of the phaseparameters p(n) of the base sequences x_(u)(n) are given as those shownin the below table.

TABLE 24 p(0), . . . , p(23) −1 −3 3 −1 −1 −1 −1 1 1 −3 3 1 3 3 1 −1 1−3 1 −3 1 1 −3 −1 −1 −3 3 −3 −3 −3 −1 −1 −3 −1 −3 3 1 3 −3 −1 3 −1 1 −13 −3 1 −1 −1 −1 1 −3 1 3 −3 1 −1 −3 −1 3 1 3 1 −1 −3 −3 −1 −1 −3 −3 −3−1 −1 −3 1 1 3 −3 1 1 −3 −1 −1 1 3 1 3 1 −1 3 1 1 −3 −1 −3 −1 −1 −3 −1−1 1 −3 −1 −1 1 −1 −3 1 1 −3 1 −3 −3 3 1 1 −1 3 −1 −1

All the 30 base sequences can be obtained by using the phase parametervalues of the 25 comparative sequences given as shown in Table 20.

FIG. 8 is a flow chart illustrating the process of a reference signaltransmission method according to an embodiment of the present invention.

Referring to FIG. 8, in step S210, the following base sequence x_(u)(n)is acquired.x _(u)(n)=e ^(jp(n)π/4)  [Equation 42]

The phase parameter p(n) is determined according to the length of thebase sequences, namely, the number of allocated resource blocks. In caseof one resource block (N=12), at least one of the 30 phase parametersp(n) given as shown in Table 17 and Table 19 can be used. In case of tworesource blocks (N=24), at least one of the 30 phase parameters p(n)given as shown in Table 20 and Table 24 can be used.

In step S220, the reference signal sequence r(n) defined by thefollowing equation by the cyclic shift ‘α’ of the base sequence x_(u)(n)is acquired.r(n)=e ^(jαn) x _(u)(n), 0≦n<N  [Equation 43]

In step S230, the reference signal sequence r(n) is mapped to a physicalresource. In this case, the physical resource may be a resource elementor a subcarrier.

In step S240, the reference signal sequence mapped to the physicalresource is converted into an SC-FDMA signal, which is then transmittedin the uplink direction.

Sequences having good correlation characteristics and CM characteristicscompared with comparative sequences are selected from among sequencesgenerated by a closed-form generating equation, and used as an uplinkreference signal. Although the sequences are used as the uplinkreference signal together with the comparative sequences, the desiredsequence characteristics can be maintained, so the data demodulationperformance can be improved and accurate uplink scheduling can bepossibly performed.

Sequences generated from a closed-form generating equation are comparedwith comparative sequences, from which those with good correlation andCM characteristics are used as an uplink reference signal. Althoughthose sequences with good correlation and CM characteristics are usedalong with the comparative sequences as the uplink reference signal,desired sequence characteristics can be maintained, to thus enhance adata demodulation performance and perform an accurate uplink scheduling.

Every function as described above can be performed by a processor suchas a microprocessor based on software coded to perform such function, aprogram code, etc., a controller, a micro-controller, an ASIC(Application Specific Integrated Circuit), or the like. Planning,developing and implementing such codes may be obvious for the skilledperson in the art based on the description of the present invention.

Although the embodiments of the present invention have been disclosedfor illustrative purposes, those skilled in the art will appreciate thatvarious modifications, additions and substitutions are possible, withoutdeparting from the scope of the invention. Accordingly, the embodimentsof the present invention are not limited to the above-describedembodiments but are defined by the claims which follow, along with theirfull scope of equivalents.

What is claimed is:
 1. A method for transmitting a reference signal in awireless communication system, the method comprising: generating areference signal sequence r(n) corresponding to a base sequence x_(u)(n)via a cyclic shift α of the base sequence x_(u)(n) as follows:r(n)=e^(jαn)x_(u)(n), where α is the cyclic shift value; andtransmitting the reference signal r(n) generated based on the basesequence x_(u)(n) on an uplink channel, wherein the base sequencex_(u)(n) is expressed by x_(u)(n)=e^(jp(n)π/4), where p(n) is a phaseparameter, and if a length of the base sequence x_(u)(n) is 12, at leastone of the sets of values provided in the below table is used as a setof values for the phase parameter p(n): p(0), . . . , p(11) −1 3 −1 1 1−3 −3 −1 −3 −3 3 −1 −1 3 1 3 1 −1 −1 3 −3 −1 −3 −1 −1 −3 1 1 1 1 3 1 −11 −3 −1 −1 3 −3 3 −1 3 3 −3 3 3 −1 −1,

or if a length of the base sequence x_(u)(n) is 24, at least one of thesets of values provided in the below table is used as a set of valuesfor the phase parameter p(n): p(0), . . . , p(23) −1 −3 3 −1 −1 −1 −1 11 −3 3 1 3 3 1 −1 1 −3 1 −3 1 1 −3 −1 −1 −3 3 −3 −3 −3 −1 −1 −3 −1 −3 31 3 −3 −1 3 −1 1 −1 3 −3 1 −1 −1 −1 1 −3 1 3 −3 1 −1 −3 −1 3 1 3 1 −1 −3−3 −1 −1 −3 −3 −3 −1 −1 −3 1 1 3 −3 1 1 −3 −1 −1 1 3 1 3 1 −1 3 1 1 −3−1 −3 −1 −1 −3 −1 −1 1 −3 −1 −1 1 −1 −3 1 1 −3 1 −3 −3 3 1 1 −1 3 −1 −1.


2. The method of claim 1, wherein the uplink channel is a physicaluplink control channel (PUCCH).
 3. The method of claim 1, wherein theuplink channel is a physical uplink shared channel (PUSCH).
 4. Themethod of claim 1, wherein the reference signal sequence r(n) is ademodulation reference signal used for demodulating uplink data.
 5. Themethod of claim 1, wherein the reference signal sequence r(n) is asounding reference signal used for user scheduling.
 6. A mobilecommunication terminal configured to transmit a reference signal in awireless communication system, the mobile communication terminalcomprising: an antenna; a communications module operatively connected tothe antenna; and a processor operatively connected to the communicationsmodule, and configured to generate a reference signal sequence r(n)corresponding to a base sequence x_(u)(n) via a cyclic shift a of thebase sequence x_(u)(n) as follows: r(n)=e^(jαn)x_(u)(n), where α is thecyclic shift value, and transmit the reference signal r(n) generatedbased on the base sequence x_(u)(n) on an uplink channel, wherein thebase sequence x_(u)(n) is expressed by x_(u)(n)=e^(jp(n)π/4), where p(n)is a phase parameter, and if a length of the base sequence x_(u)(n) is12, at least one of the sets of values provided in the below table isused as a set of values for the phase parameter p(n): p(0), . . . ,p(11) −1 3 −1 1 1 −3 −3 −1 −3 −3 3 −1 −1 3 1 3 1 −1 −1 3 −3 −1 −3 −1 −1−3 1 1 1 1 3 1 −1 1 −3 −1 −1 3 −3 3 −1 3 3 − 3 3 3 −1 − 1,

or if a length of the base sequence x_(u)(n) is 24, at least one of thesets of values provided in the below table is used as a set of valuesfor the phase parameter p(n): p(0), . . . , p(23) −1 −3 3 −1 −1 −1 −1 11 −3 3 1 3 3 1 −1 1 −3 1 −3 1 1 −3 −1 −1 −3 3 −3 −3 −3 −1 −1 −3 −1 −3 31 3 −3 −1 3 −1 1 −1 3 −3 1 −1 −1 −1 1 −3 1 3 −3 1 −1 −3 −1 3 1 3 1 −1 −3−3 −1 −1 −3 −3 −3 −1 −1 −3 1 1 3 −3 1 1 −3 −1 −1 1 3 1 3 1 −1 3 1 1 −3−1 −3 −1 −1 −3 −1 −1 1 −3 −1 −1 1 −1 −3 1 1 −3 1 −3 −3 3 1 1 −1 3 −1 −1.


7. The mobile communication terminal of claim 6, wherein the uplinkchannel is a physical uplink control channel (PUCCH).
 8. The mobilecommunication terminal of claim 6, wherein the uplink channel is aphysical uplink shared channel (PUSCH).
 9. The mobile communicationterminal of claim 6, wherein the reference signal sequence r(n) is ademodulation reference signal used for demodulating uplink data.
 10. Themobile communication terminal of claim 6, wherein the reference signalsequence r(n) is a sounding reference signal used for user scheduling.